college algebra tutorial

College Algebra TutorialI. Concepts of Algebra1. Exponents and Radicals2. Factoring3. Fractional Expressions4. Graphical Representation of DataII. Equations and Inequalities1. Linear equations2. Quadratic Equations3. Complex Numbers4. Other Types of Equations5. Linear Inequalities6. Other Types of InequalitiesIII. Functions and Graphs1. Lines2. Function Concepts3. Functions and Graphs4. Shifting Graphs5. Combining Functions6. Inverse FunctionsIV. Exponential and Logarithmic Functions1. Exponential Functions2. Logarithmic Functions3. Laws of Logarithm4. Exponential and Logarithmic Equations

Exponents and RadicalsI. Simplify each of thefollowing expressions.II Radicals

5. 1.7. Find the value of

2. 8. Simplify by removing all possible factors from the radical.



5. 11. Rationalize the denominator and simplify.

6. 12. Rationalize the denominator and simplify.

Simplify the expression

Commute the terms so as to group the constant terms together andgroup the variable terms together.

Simplify by using the first law of exponents.

Simplify the expression x3(2y z2)3x3(2y z2)3

= x3 . 23 y3 (z2)3Distribute the exponent of 3 among the three factors in (2y z2)3 using the properties of exponents.

= x3 . 8y3z6= 8 x3 y3 z6Simplify by using the properties of exponents andwrite your answer in standard form.


Use the commutative law to rearrange the factors of the numerator and simplify the denominator using one of the properties of exponents.

Simplify the numerator by multiplying the constant term and combining the variable terms using the properties of exponents.

Combine the variable terms found in the numerator and denominator using the properties of exponents.


Simplify the second factor in the numerator.

Simplify the numerator further by combining the variable termsand the constant terms.

Combine the variable terms found in the numerator and denominator.

Write this expression without negative exponents.


Distribute the -3 exponent among the three factors in parentheses using the properties of exponents.

Simplify each of the three terms using the properties of exponents.

Simplify and write this expression with no negative exponents.


Distribute the fractional exponent among the factors in parentheses using properties of exponents.

Simplify each of these three factors using properties of exponents

Continue to simplify each factor.

Write this expression without negative exponents.

Find

4Try to think of a number whose cube is 64

Simplify by removing all possible factors from the radical

Using properties of radicals, rewrite the single radical as a product of three radicals.

Simplify each radical where possible.


Rewrite each of the 3 factors under the radical as a product so that you can take the cube root of one of the factors.

Rewrite the expression as a product of radicals.

Find the cube root of each factor where possible.


Write the expression as a quotient of two radicals, and factor the numerator in preparation to extract a radical.

Write the numerator as a product of two radicals.

Simplify by extracting square roots where possible.

Rationalize the denominator and simplify

You are to write the given fraction as an equivalent fraction where the new fraction has no radical in the denominator.

Simplify the denominator.

Rationalize the denominator and simplify

Multiply the numerator and denominator by the conjugate of the denominator.

Multiply the two binomials in the denominator.i.e. FOIL the denominator.


Continue simplifying the denominator.

Reduce the constant coefficients in this fractional expression.

Fractional ExpressionsSimplify the rational expressions and write them in reduced form.10.

1.11.

2.12.

3.13.

4.14.

Perform the operation and write the result in reduced form.15.

5.16.

6.17.

7.18.

8.19.

9.

Simplify the rational expression

and write it in reduced form.Factor a common factor from the two terms in the denominator.

Divide a common factor from the numerator and denominator.

Write the expression in reduced form.


and write it in reduced form.Factor the numerator. Note that the numeratorhas four terms. This should suggest a factoringmethod.

Continue factoring the numerator by grouping.

Note that 2 - a is the negative of a - 2.Reduce this fraction by dividing a - 2 into thenumerator and into the denominator.

Write this rational expression in simplified reduced form.


and write it in reduced form.Notice that the numerator is a sumof two cubes. Write it as such.

Factor the numerator using the formulafor factoring the sum of two cubes.

Divide out the common factor found in thenumerator and in the denominator.

= (a + 2)Write this rational expression in simplified reduced form.

Perform the multiplication

and write the results in reduced form.Perform the operation by multiplying the numerators together and multiplying the denominators together.

Factor the difference of two squares whichis found in the denominator.

Divide out the common factor found in thenumerator and in the denominator.

Continue to reduce this fractional expression bydividing the numerator and denominator by a.

Write this rational expression in simplified reduced form.



Completely factor the numerator andcompletely factor the denominator

Divide out the common factors found in thenumerator and in the denominator.

= (a + 3 )Write the expression in simplified reduced form.



Factor the numerator and denominator.

Notice that the numerator contains a factor which is a difference of two cubes. Write this factor as a difference of two cubes.

Factor the difference of two cubes found in the numerator.

Divide out terms common to the numerator and denominator.



Factor the numerator and denominator.



Perform the division

and write the results in reduced form.Perform the operation by multiplying the numerator by the reciprocal of the denominator.

Factor the numerator and denominator. Note the numerator contains a difference of two squares.

Factor the numerator further.



Perform the division

and write the results in reduced form.Perform the operation by multiplying the numerator by the reciprocal of the denominator.

Factor the numerator and denominator. Note the denominator contains a difference of two cubes and a sum of two cubes.

Factor the denominator further.



Perform the addition

and write the results in reduced form.Factor the denominators into prime factors so as to assist in finding the least common denominator.

2.52.3 = 150What is the least common denominator?

Write each fraction in an equivalent form with 150 as the common denominator.

Perform the addition.

Perform the subtraction

and write the results in reduced form.(2x + 1)2Find the "least" common denominator for these two fractions.

Write each fraction in an equivalent form with (2x + 1)2 as the common denominator.


Simplify the expression.


and write the results in reduced form.(x + 2)Find the "least" common denominator for these two fractions.

Write each fraction in an equivalent form with (x + 2) as the common denominator.




and write the results in reduced form.Notice that (2 - x) = - (x - 2). Rewrite the denominator of the second fraction using this equivalence.

Write each fraction in an equivalent form with (x - 2) as the common denominator.

Perform the addition.



and write the results in reduced form.Factor each denominator to assist in finding the "least" common denominator.

(x - 1)(x - 2)(x + 2)Using the information above, what is the "least" common denominator?

Write each fraction in an equivalent form with (x - 1)(x - 2)(x + 2) as the common denominator.

Perform the subtraction.



and write the results in reduced form.Factor each denominator to assist in finding the "least" common denominator.

(y + 2)2 (y-1)Using the information above, find the "least" common denominator.

Write each fraction in an equivalent form with (y + 2)2(y-1) as the common denominator.

Perform addition.


Perform the operations

and write the result in reduced form.3The two terms of the numerator first need to be combined into a single fractional expression. What is the least common denominator of the two terms in the numerator?

Write each fraction in the numerator as an equivalent fraction with 3 as the common denominator.

Combine the two terms of the numerator into a single fraction.

Multiply the numerator and denominator by the reciprocal of the denominator.

Simplify this expression.

Perform the operation

and write the result in reduced form.y ( y + 1)The two terms of the numerator first need to be combined into a single fractional expression. What is the least common denominator of the two terms in the numerator?

Write each fraction in the numerator as an equivalent fraction with y(y + 1)as the common denominator.

Combine the two terms of the numerator into a single fraction.

Simplify the fraction in the numerator.

Multiply the numerator and denominator by the reciprocal of the denominator.


Graphical Representation of Data1. Determine the quadrant in which (x , y) is located so that x < 0 and y > 0. 2. Determine the quadrant in which (x , y) is located so that - x < 0 and y < 0. 3. Determine the quadrants in which (x , y) is located so that x < -4 . 4. a) Sketch the line segment with endpoints at (-2,3) to (4,7).b) Find the midpoint of the segment with endpoints at (-2,3) to (4,7). c) Find the length of segment with endpoints at (-2,3) to (4,7). 5. a) Plot the triangle with vertices at A( 8, 5 ), B( 1, -2 ) and C( -2, 2 ).b) Show this is a right triangle by :i) finding the length of each of the three sides andii) showing that these three lengths satisfies the Pythagorean Theorem.Graphical Representation of Data1. Determine the quadrant in which (x , y) is located so that x < 0 and y > 0. Sketch the part of the (x,y) plane where x < 0.

Sketch the part of the plane where y > 0.

The intersection of the above two shaded areaswill give the part of the plane where the x < 0 and y > 0. Find this intersection by first overlayingthe two shaded areas.

Quadrant IIThe part of the plance where x < 0 and y > 0 is thecrossed hatched area. Which quadrant is this?

Graphical Representation of Data2. Determine the quadrant in which (x , y) is located so that - x < 0 and y < 0. (-1)(-x) > (-1) 0x > 0In order to more easily picture the part of the planewhere -x < 0, multiply both sides of this inequalityby -1. Remember to switch the direction of the inequality

Sketch the part of the plane where x > 0.

Sketch the part of the plane where y < 0

The intersection of the above two shaded areaswill give the part of the plane where the x > 0 and y < 0. Find this intersection by first overlayingthe two shaded areas.

Quadrant IVThe part of the plance where x > 0 and y < 0 is thecrossed hatched area. Which quadrant is this?

Graphical Representation of Data3. Determine the quadrants in which (x , y) is located so that x < -4 . Sketch the part of the (x,y) plane where x < -4.

The points (x,y) where x < -4lie in either quadrant II or III.In which quadrants do the points (x,y) lie so thatx < -4.

Graphical Representation of Data4. a) Sketch the line segment with endpoints at (-2,3) to (4,7).b) Find the midpoint of the segment with endpoints at (-2,3) to (4,7). c) Find the length of segment with endpoints at (-2,3) to (4,7). Sketch the line segment with endpoints at (-2,3) to (4,7).

Recall the midpoint formula for a line segment.

Substitute into the midpoint formula in order to find the midpoint of the segment from (-2,3) to (5,7). Write your answer in simplified form.

Place this midpoint (1,5) on the segment and check thatit visually looks like the midpoint of the segment.

Recall the formula for the distance between two points.

Substitute into this formula to find the distance from (-2,3) to (5,7).

Graphical Representation of Data5. a) Plot the triangle with vertices at A( 8, 5 ), B( 1, -2 ) and C( -2, 2 ).b) Show this is a right triangle by :i) finding the length of each of the three sides andii) showing that these three lengths satisfies the Pythagorean Theorem.Plot the three points A( 8, 5 ), B( 1, -2 ) and C( -3, 2 ).

Find the length of side ABby using the distance formula.

Find the lengths of side BC and CA.

Check whether the lenghts satisfies the criteria for thePythagorean Theorem, i.e. check whetherAB2 + BC2 = AC2.

Linear Equations1. Determine which of the x values are solutions of the equation x4 + x3 - 5x2 + x - 6 = 0.a) x = -3, b) x = -2, c) x = 1, d) x = 22. Determine whether the equation - 5(x+2) +2 = -3x + 5 is true for all values of x (and hence an identity) or if is true for only some values of x (and hence a conditional equation.)3. Determine whether the equation - 4(x-3) + 7x = 3(x+4) is true for all values of x (and hence an identity) or if is true for only some values of x (and hence a conditional equation.)Solve the equation if possible and check your solution:4. 7x - 1 = 3(x + 5)5. 5[x - (3x + 2)] = 9 - 5x6. 1.4x - .8 = .3(4 - 5x)7. 8. 9. 10. 11. 12. Solve for x:13. 5(x+2) + ax = 5 - x14. 6 - bx = 15 + cx 15. Determine which of the x values are solutions to the equation

a) x = -3, b) x = -2, c) x = 1 , d) x = 2.Substitute -3 into the equation for x and determine if the equation is satisfied.

Evaluate this expression and determine if the two sides are equal.

Yes, since both sides are 0.Is x = -3 a solution?

Substitute -2 into the equation for x and determine if the equation is satisfied.


No, since the two sides are unequal.Is x = -2 a solution?

Substitute 1 into the equation for x and determine if the equation is satisfied.


No, since the two sides are unequal.Is x = 1 a solution?

Substitute 2 into the equation for x and determine if the equation is satisfied.


Yes, since both sides are 0Is x = 2 a solution?

Determine whether the equation - 5 (x + 2 ) + 2 = -3x + 5 is true for all values of x (and hence an identity) or if it is true for only some values of x (and hence a conditional equation.)- 5 (x + 2 ) + 2= -5x -10 + 2= -5x - 8Simplify the left hand side and determine if it is identical to the right hand side

No, because the simplified left side is not equal to the right hand side.Is this an identity?

- 5 (x + 2 ) + 2 = -3x + 5Find the values for which this conditional equation is true.

-5x - 8 = -3x + 5Simplify the left hand side.

-5x + 3x = 5 + 8Perform operations to write an equivalent equation with the variables on the left and the constants on the right.

-2x = 13x = - (13 / 2)Simplify and solve.

Determine whether the equation -4 (x-3) + 7x = 3(x+4)is true for all values of x (and hence an identity) or if it is true for only some values of x (and hence a conditional equation.)-4 (x-3) + 7x= -4x + 12 +7x= 3x + 12= 3(x + 4)Simplify the left hand side and determine if it is identical to the right hand side

Yes, because the left side is shown to be equal to the right hand side.Is this an identity?

Solve the equation 7x - 1 = 3 (x + 5)and check your answer7x - 1 = 3 (x + 5)7x - 1 = 3x + 15Apply the distributive law to the right hand side.

7x - 3x = 15 + 1Transform the equation to one with the variables on the left and the constants on the right.

4x = 16x = 4Simplify each side and solve for x.

7(4) - 1 = ? 3(4 + 5)28 -1 ?= 3(9)27 = 27Yes, x = 4 is a solution.Check your answer by substituting x = 4 into the equation.

Solve the equation 5 [x - (3x + 2)] = 9 - 5xand check your answer5 [x - (3x + 2)] = 9 - 5x5[x -3x-2] = 9 - 5x5[-2x-2] = 9 - 5x-10x -10 = 9 - 5xSimplify the left hand side.

-10x + 5x = 9 + 10Transform the equation to one with the variables on the left and the constants on the right.

-5x = 19x = -(19 / 5) Simplify each side and solve for x.

- 19 + 3 (19) - 10 =? 9 + 1928 = 28 Yes, x = -(19 / 5)is a solution.Check your answer by substituting x = -(19 / 5) into the equation.

Solve the equation 1.4x - .8 = .3(4 - 5x)and check your answer1.4x - .8 = .3(4 - 5x)1.4x - .8 = 1.2 - 1.5xApply the distributive law to the right hand side.

(10) [1.4x - .8 ] = (10) [1.2 - 1.5x]14 x - 8 = 12 - 15xMultiply both sides by 10 so as to create an equivalent equation with no decimals in it.

14x + 15x = 12 + 8Transform the equation to one with the variables on the left and the constants on the right.

29x = 20

Simplify each side and solve for x.

Check your answer by substituting

Solve the equation

Multiply both sides by the least common denominator to create an equivalent equation with no denominators.

10 - 15 x = 2xSimplify both sides.

Solve for x.

Solve the equation

Multiply both sides by the least common denominator to create an equivalent equation with no denominators.

Simplify both sides.

12x + 8 = 6(x + 5)12x + 8 = 6x + 30 Continue simplifying both sides.

12x - 6x = 30 - 8Transform the equation to one with the variables on the left and the constants on the right.

6x = 22


Solve the equation

Factor the denominator on the right hand side so as to help in determining a "least" common denominator.

Multiply both sides by the "least" common denominator to create an equivalent equation with no denominators.

Apply the distributive law on the left side and perform the multiplication on the right.

3(x-3) + (x+2) = 4 Reduce each fractional expression.

3x - 9 + x + 2 = 44x - 7 = 44x = 11


Since x = 11 / 4 is not a zero of any of the denominators, it is in fact a valid solution.Is this answer an extraneous solution or a valid solution?

Solve the equation

Factor the denominator on the left hand side so as to help in determining a "least" common denominator.



7 - 4(x - 2) = 5 (x + 2)Reduce each fractional expression.

7 - 4x + 8 = 5x + 1015 - 4x = 5x + 10-9x = -5


Since x = 5 / 9 is not a zero of any of the denominators, it is in fact a valid solution.Is this answer an extraneous solution or a valid solution?

Solve the equation


Apply the distributive law on the right side and perform the multiplication on the left.

9y = 3 + 2(3y - 1)Reduce each fractional expression.

9y = 3 + 6y - 23y = 1

Simplify each side and solve for y.

Since x = 1/3 is a zero of two of the denominators, these fractions are undefined at x = 1/3 . Hence this is an extraneous solution.Is this answer an extraneous solution or a valid solution?

Solve the equation

Factor the denominator on the right. hand side so as to help in determining a "least" common denominator.



y + 2 + 4 (y - 2) = 5y - 6Reduce each fractional expression.

y + 2 + 4y - 8 = 5y - 65y - 6 = 5y - 6Simplify each side and solve for y.

Since this equation is an identity, it is true for all values of y. However, in determining the solution to the original equation, we must exclude all values of y where the denominators are zero. Hence, the solution set is all values of y except y = 2 and y = -2.We note that the above equation is an identity. How do we interpret this fact with regard to the solution set to the original equation?

Solve the equation

for x.Strategy: Transform the equation to one with the variable terms on one side and the constant terms on the other. Factor out x from the side containing it and then solve for x.

5x + 10 + ax = 5 - xApply the distributive law on the left hand side.

5x + ax + x = 5 - 10Write an equivalent equation with all variables on the left and the constant terms on the right.

6x + ax = - 5(6 + a) x = - 5

Simplify, factor out an x and then solve for x.

Solve the equation

for x.Strategy: Transform the equation to one with the variable terms on one side and the constant terms on the other. Factor out x from the side containing it and then solve for x.

6 - 15 = cx + bxWrite an equivalent equation with all variables on the right and the constant terms on the left.

-9 = (c + b) x

Simplify, factor out an x and then solve for x.

Solve the equation

for x.Strategy:1. Rewrite the equation as an equation with no denominator.2. Transform the equation to one with the terms involving xon one side and all remaining terms on the other side. Factor out x from the side containing it and then solve for x.

y(2-3x) = 2xRewrite the equation as an equation with no denominator by multiplying both sides by 2 - 3x.

2y - 3yx = 2xExpand the left hand side by using the distributive law.

2y = 2x + 3yxWrite an equivalent equation with terms involving x on the right and all remaining terms on the left.

2y = (2 + 3y) x

Factor out an x and then solve for x.

Quadratic EquationsWrite the quadratic equation in standard form.1. 6x = 5 - 2x22. 9 = 2(2x +4)2Solve the equation by factoring.3. 2y2 - 5y -3 = 04. 12 - 15x2 = 8x5. 6. y2 - 8ay - 9a2 = 0Solve the equation by extracting square roots.7. 4y2 = 258. (y - 3)2 = 10Solve the equation by completing the square.9. y2 + 6y + 7 = 010. 9y2 - 36y = -5Solve the equation by using the quadratic formula.11. 3x2 + 2x -1 = 012. 4y - 2 = y213. Write the quadratic equation in standard form.6x = 5 - 2x26x = 5 - 2x22x2 + 6x - 5 = 0Rewrite the equation with all terms on one side of the equal sign, with the x2 term first, with the x term in the middle and the constant term last

Write the quadratic equation 9 = 2(2x + 4)2in standard form.9 = 2(2x + 4)29 = 2(4x2 + 16x + 16)9 = 8x2 + 32x + 32Expand the expression on the right hand side of the equation.

0 = 8x2 + 32x + 23 or8x2 + 32x + 23 = 0Rewrite the equation with all terms on one side of the equal sign, with the x2 term first, with the x term in the middle and the constant term last.

Solve the equation 2y2 - 5y - 3 = 0by factoring.2y2 - 5y - 3 = 0 (2y + 1)(y - 3) = 0 Factor using the usual factoring method for trinomials.

2y + 1 = 0 ; y - 3 = 0 This product will be zero if either (or both) factors are zero. Set each factor equal to zero.

y = - (1/2) ; y = 3 Solve each of these two linear equations.

Solve the equation 12 - 15x2 = 8xby factoring.12 - 15x2 = 8x 0 = 15x2 + 8x -12 or15x2 + 8x -12 = 0 Write the equation in standard form.

(5x+ 6)(3x - 2) = 0Factor using the usual factoring method for trinomials.

5x+ 6 = 0 ; 3x - 2 = 0 This product will be zero if either (or both) factors are zero. Set each factor equal to zero.

x= - ( 6/5 ) ; x = 2/3 Solve each of these two linear equations.

Solve the equation

by factoring.Strategy:1. Factor the denominator.2. Multiply both sides by the least common denominator.3. Write the equation in standard quadratic form.4. Factor and solve the equation.

Factor the denominator on the right hand side.

Multiply both sides by the "least" common denominator.

Use the distributive law and perform the multiplication.

2x2 + 5(x + 3) - 4x2 - 12x = 182x2 + 5x + 15 - 4x2 - 12x = 18-2x2 -7x - 3 = 02x2 + 7x + 3 = 0Simplify each fractional expression and then write the equation in standard quadratic form.

(2x + 1)(x + 3) = 0Factor the equation.

2x + 1 = 0; x + 3 = 0x = - (1/2) ; x = - 3Set each factor equal to zero and then solve each of these two linear equations.

Solve the equation y2 - 8ay - 9a2 = 0for y by factoring.(y - 9a)(y + a) = 0 Factor using the usual factoring method for trinomials.

y - 9a = 0 ; y + a = 0 This product will be zero if either (or both) factors are zero. Set each factor equal to zero.

y = 9a ; y = -a Solve each of these two linear equations for y.

Solve the equation 4y 2 = 25by extracting square roots.Divide both sides by 4.

Apply the square root operator to both sides and simplify..

Solve the equation (y - 3)2 = 10for y by extracting square roots.Apply the square root operator to both sides the equation.

Solve for y .

Solve the equation y2 + 6y + 7 = 0by completing the square.Strategy:1. Rewrite the equation with just the variable terms on the left.2. Complete the square on the left side.3. Extract square roots from both sides.4.Solve the equation.

y2 + 6y = -7 Rewrite the equation with just the variable terms on the left.

y2 + 6y + ___= -7 + ___Add the same number to both sides so as to make the left side a perfect square.

y2 + 6y + 32= -7 + 32This is done by taking half the coefficient of the linear term (i.e. half of 6), squaring that number and adding it to both sides.

(y + 3) 2 = -7 + 9(y + 3)2 = 2 Write the left side as a perfect square and simplify the right side.

Extract square roots from both sides and solve for y.

Solve the equation 9y2 - 36y = -5by completing the square.Strategy:1. Rewrite the equation with just the variable terms on the left.2. Complete the square on the left side.3. Extract square roots from both sides.4.Solve the equation.

9(y2 - 4y) = -59(y2 - 4y + ___) = -5 + ___Rewrite the equation with just the variable terms on the left. Add the same number to both sides so as to make the left side a perfect square.

9(y2 - 4y + 22 ) = -5 + 9.22This is done by taking half the coefficient of the linear term (i.e. half of 4), squaring that number, multiply it by the coefficient 9 and then adding that product to both sides.

9(y - 2) 2 = -5 + 36Write the quadratic polynomial on the left side as a perfect square and simplify the right side.

9(y - 2) 2 = 31 Simplify the right hand side.

Divide both sides by 9.

Extract square roots from both sides.

Solve for y.

Solve the equation

using the quadratic formula.5y = -1(y2 -9) 5y = -y2 + 9 Rewrite this equation without a denominator

y2 + 5y - 9 = 0Write the equation in standard form.

Recall the quadratic formula.

a = 1; b = 5 c = -9Write the values of a, b, c.

Substitute the values of a, b, c, into the quadratic formula.


Express the two solutions in simplified form.

Complex NumbersI. Write the complex number in standard form (i.e. in the form a + b i.)1. 2. -5i + 3 i 2II. Perform the operation and write the result in standard form.3. (2 + 3i) - (6 -4i)4. 5. (2 + 3i)(1 - 5i)6. 7i(7 - 3i)7. (3 + 2i)2 + (4 - 3i)III. Write the conjugate of the complex number.8. 3 + 5i9. 2 - 6i10. 15iIV. Perform the operation and write the result in standard form.11. 12. 13. V. Solve the quadratic equations using the Quadratic Formula.14. 3x2 + 9x +7 = 015. y2 -2y + 2 = 0Write the complex number

in standard form

Write this expression in standard form.

Write the complex number-5i + 3 i 2in standard form-5i + 3 i 2= -5i + 3 (-1)Write i 2 as a real number.

= -3 - 5iWrite this expression in standard form.

Perform the operation and(2+3i) - (6 - 4i)write the result in standard form(2 + 3i) - (6 - 4i) = 2 + 3i - 6 + 4iRewrite the expression without parentheses.

= 2 - 6 + 3i + 4i= -4 + 7iCombine the real parts and combine the imaginary parts.Write the result in standard form.

Perform the operation and

write the result in standard formSimplify the radicals and express them using complex notation.

Combine the real parts and combine the imaginary parts.Write the result in standard form.

Perform the operation and(2 + 3i) (1 - 5i)write the result in standard form(2 + 3i) (1 - 5i)= 2 - 10 i + 3i - 15 i 2= 2 - 10 i + 3i - 15(-1)Expand the product, and simplify.

= 2 + 15 - 10 i + 3i= 17 - 7iCombine the real parts and combine the imaginary parts.Write the result in standard form.

Perform the operation and7i (7 - 3i)write the result in standard form7i (7 - 3i)= 49i - 21i2= 49i - 21(-1)Expand the product and simplify.

= 21 + 49 iWrite the result in standard form.

Perform the operation and(3 + 2i) 2 + (4 - 3i)write the result in standard form(3 + 2i) 2 + (4 - 3i)= (9 + 6i + 6i + 4i2) + (4 - 3i)= (9 + 12i + 4(-1)) + (4 - 3i)Expand the squared term and simplify

= 9 -4 + 4 +12i - 3i= 9 + 9i Group the imaginary terms together,group the real terms together.Write the result in standard form.

Write the complex conjugate of 3 + 5iand express the result in standard form3 + 5i Conjugate is 3 - 5iWrite the complex conjugate.

Write the complex conjugate of 2 - 6iand express the result in standard form2 - 6iConjugate is 2 + 6iWrite the complex conjugate.

Write the complex conjugate of 15iand express the result in standard form15iConjugate is - 15iWrite the complex conjugate.


write the result in standard formMultiply the numerator and denominator by the conjugate of the denominator of the original quotient.

Perform the multiplication of the two fractions.

Perform the multiplication in the numerator and perform the multiplication in the denominator.


Write the result in standard form.


write the result in standard formMultiply the numerator and denominator by the conjugate of the denominator of the original quotient.

Perform the multiplication of the two fractions.

Perform the multiplication in the numerator and perform the multiplication in the denominator.

Simplify the numerator and denominator.



write the result in standard form(3 + 2i)(3 -2i)Name the common denominator to be used to combine the two fractions into one fraction.

Write each fraction as an equivalent fraction with this common denominator.

Combine the two fractions into one fraction.

Simplify the numerator and the denominator.


Solve the quadratic equation3x2 + 9x + 7 = 0using the quadratic formula.Recall the quadratic formula

Substitute the values for a, b, and c into this formula.

Simplify the expression under the radical

Rewrite this expression using complex number notation.

Write the two answers in standard form.

Solve the quadratic equationy2 - 2y + 2= 0using the quadratic formula.Recall the quadratic formula

Substitute the values for a, b, and c into this formula.

Simplify the expression under the radical

Rewrite this expression using complex number notation.

Write the two answers in standard form.

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