UNIT 7 STUDY SHEET – QUADRATIC EQUATIONS

METHODS USED TO SOLVE QUADRATIC EQUATIONS

1) Factoring Get the equation into standard form {ax2 + bx + c = 0}.

Factor the side of the equation not equal to zero.

Set each factor containing the variable equal to zero.

Solve each of the resulting equations.

Check by substituting each root (separately) in to the original equation.

Factoring with GCF (greatest common factor)

Factoring with DOTS (difference of two squares)

Factoring Trinomials

Find the largest value that can

be factored from each of the

elements of the expression.

Look carefully at this example to

refresh this process:

Factoring Harder

Trinomials (AC method)

If the leading coefficient is not

equal to 1, you must think more

carefully about how to set up

your factors.

In a quadratic equation in

descending order with a leading

coefficient of one, look for the

product of the roots to be the

constant tern and the sum of the

roots to be the coefficient of the

middle term.

2) Square Root Method Use for equations in which a perfect square is equal to a constant.

First isolate the perfect square

Then take the square-root of both sides (remember the )

Solve each of the resulting equations for x

3) Completing the Square

Steps for completing the square:

1. Be sure that the coefficient of the highest power is one. If it is not, divide each term by that value to create a leading coefficient of one.

2. Move the constant term to the right hand side.

3. Prepare to add the needed value to create the perfect square trinomial. Be sure to balance the equation. The boxes may help you remember to balance.

4. To find the needed value for the perfect square trinomial, take half of the coefficient of the middle term (x-term), square it, and add that value to both sides of the equation.

5. Factor the perfect square trinomial.

6. Take the square root of each side and solve. Remember to consider both plus and minus results.

4) The Quadratic Formula {for equations written in the form ax2 + bx + c = 0}

a2

ac4bbx

2

Solving Quadratic Equations

Method Can be Used When to Use

Factoring Sometimes

Use if the constant term is 0 or if the factors are easily determined. Example: x2 – 7x = 0

Square Root Method Sometimes

Use for equations in which a perfect square is equal to a constant. Example: (x – 5)2 = 18

Completing the Square Always

Useful for equations of the form x2 + bx + c = 0, where b is even. Example: x2 + 6x – 14 = 0

Quadratic Formula Always

Useful when other methods fail or are too tedious. Example: 2.3x2 – 1.8x + 9.7 = 0

5) Graphing Set quadratic equation equal to zero. On calculator, type equation into Y = and graph. Roots of the equation are the x-intercepts.

Calculator Information – Finding the Zeros of a function (x-intercepts)

Solving a System of Equations Algebraically:

Solve the linear equation for one variable. Substitute the linear equation into the quadratic equation and solve. Find the y-values by substituting each value of x into the linear equation. Check each solution in both equations to check for extraneous roots.

Solving a System of Equations Graphically:

Solve each equation for y. Graph and label each equation on the same set of axes. {You may use your graphing calculator

to help.} The point(s) of intersection are the solutions to the system. If there are no points of intersection, then there is no solution to the system of equations. Check each solution in both equations. Please note: If you are graphing a circle, you can identify the center and radius of the circle and

use a compass to draw.

Circles:

Circle whose center is at (h,k) (This will be referred to as the "center-radius form".

It may also be referred to as "standard form".)

Equation: Example: Circle with center (2,-5), radius 3

Convert Circles to Center-Radius Form by Completing the Square: