holt algebra 1 9-9 the quadratic formula and the discriminant 9-9 the quadratic formula and the...
TRANSCRIPT
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant9-9 The Quadratic Formula
and the Discriminant
Holt Algebra 1
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Warmup
Solve using square roots. Check your answers.
1. x2 – 195 = 1
2. 4x2 – 18 = –9
3. 2x2 – 10 = –12
4. Solve 0 = –5x2 + 225. Round to the nearest hundredth.
± 14
± 6.71
no real solutions
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Lesson Quiz: Part II
1. A community swimming pool is in the shape of a trapezoid. The height of the trapezoid is twice as long as the shorter base and the longer base is twice as long as the height.
The area of the pool is 3675 square feet. What is the length of the longer base? Round to the nearest foot.
(Hint: Use )
108 feet
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Warm Up
Evaluate for x =–2, y = 3, and z = –1.
6 1. x2 2. xyz
3. x2 – yz 4. y – xz
4
5. –x 6. z2 – xy
7 1
7 2
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Solve quadratic equations by using the Quadratic Formula.
Determine the number of solutions of a quadratic equation by using the discriminant.
Objectives
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
discriminant
Vocabulary
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax2 + bx + c = 0, you can derive the Quadratic Formula. The Quadratic Formula is the only method that can be used to solve any quadratic equation.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
To add fractions, you need a common denominator.
Remember!
Holt Algebra 1
9-9 The Quadratic Formula and the DiscriminantExample 1A: Using the Quadratic Formula
Solve using the Quadratic Formula.
6x2 + 5x – 4 = 0
6x2 + 5x + (–4) = 0 Identify a, b, and c.
Use the Quadratic Formula.
Simplify.
Substitute 6 for a, 5 for b, and –4 for c.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Example 1A Continued
Solve using the Quadratic Formula.
6x2 + 5x – 4 = 0
Simplify.
Write as two equations.
Solve each equation.
Holt Algebra 1
9-9 The Quadratic Formula and the DiscriminantExample 1B: Using the Quadratic Formula
Solve using the Quadratic Formula.
x2 = x + 20
1x2 + (–1x) + (–20) = 0 Write in standard form. Identify a, b, and c.
Use the quadratic formula.
Simplify.
Substitute 1 for a, –1 for b, and –20 for c.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Example 1B Continued
Solve using the Quadratic Formula.
x = 5 or x = –4
Simplify.
Write as two equations.
Solve each equation.
x2 = x + 20
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
You can graph the related quadratic function to see if your solutions are reasonable.
Helpful Hint
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Many quadratic equations can be solved by graphing, factoring, taking the square root, or completing the square. Some cannot be solved by any of these methods, but you can always use the Quadratic Formula to solve any quadratic equation.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Example 2: Using the Quadratic Formula to Estimate Solutions
Solve x2 + 3x – 7 = 0 using the Quadratic Formula.
Use a calculator: x ≈ 1.54 or x ≈ –4.54.
Check reasonableness
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
If the quadratic equation is in standard form, the discriminant of a quadratic equation is b2 – 4ac, the part of the equation under the radical sign. Recall that quadratic equations can have two, one, or no real solutions. You can determine the number of solutions of a quadratic equation by evaluating its discriminant.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
3x2 – 2x + 2 = 0 2x2 + 11x + 12 = 0 x2 + 8x + 16 = 0
a = 3, b = –2, c = 2 a = 2, b = 11, c = 12 a = 1, b = 8, c = 16
b2 – 4ac b2 – 4ac b2 – 4ac
(–2)2 – 4(3)(2) 112 – 4(2)(12) 82 – 4(1)(16)
4 – 24 121 – 96 64 – 64
–20 25 0
b2 – 4ac is negative.
There are no real solutions
b2 – 4ac is positive.
There are two real solutions
b2 – 4ac is zero.
There is one real solution
Example 3: Using the DiscriminantFind the number of solutions of each equation using the discriminant.
A. B. C.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t2 + vt + c, where c is the beginning height of the object above the ground.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
If the object is shot straight up from the ground, the initial height of the object above the ground equals 0.
Helpful Hint
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t2 + vt + c, where c is the initial height of the object above the ground. The ringer on a carnival strength test is 2 feet off the ground and is shot upward with an initial velocity of 30 feet per second. Will it reach a height of 20 feet? Use the discriminant to explain your answer.
Example 4: Application
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Example 4 Continued
h = –16t2 + vt + c
20 = –16t2 + 30t + 2
0 = –16t2 + 30t + (–18)
b2 – 4ac
302 – 4(–16)(–18) = –252
Substitute 20 for h, 30 for v, and 2 for c.
Subtract 20 from both sides.
Evaluate the discriminant.
Substitute –16 for a, 30 for b, and –18 for c.
The discriminant is negative, so there are no real solutions. The ringer will not reach a height of 20 feet.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
There is no one correct way to solve a quadratic equation. Many quadratic equations can be solved using several different methods.
Holt Algebra 1
9-9 The Quadratic Formula and the DiscriminantExample 5: Solving Using Different Methods
Solve x2 – 9x + 20 = 0. Show your work.
Method 1 Solve by graphing.
y = x2 – 9x + 20Write the related quadratic
function and graph it.
The solutions are the x-intercepts, 4 and 5.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Solve x2 – 9x + 20 = 0. Show your work.
Method 2 Solve by factoring.
x2 – 9x + 20 = 0
Example 5 Continued
(x – 5)(x – 4) = 0
x – 5 = 0 or x – 4 = 0
x = 5 or x = 4
Factor.
Use the Zero Product Property.
Solve each equation.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Solve x2 – 9x + 20 = 0. Show your work.
Method 3 Solve by completing the square.
Example 5 Continued
x2 – 9x + 20 = 0
Add to both sides.
x2 – 9x = –20
x2 – 9x + = –20 +
Factor and simplify.
Take the square root of both sides.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Solve x2 – 9x + 20 = 0. Show your work.
Method 3 Solve by completing the square.
Example 5 Continued
Solve each equation.
x = 5 or x = 4
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Method 4 Solve using the Quadratic Formula.
Example 5: Solving Using Different Methods.
1x2 – 9x + 20 = 0
x = 5 or x = 4
Identify a, b, c.
Substitute 1 for a, –9 for b, and 20 for c.
Simplify.
Write as two equations.
Solve each equation.
Solve x2 – 9x + 20 = 0. Show your work.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Notice that all of the methods in Example 5 (pp. 655-656) produce the same solutions, –1 and –6. The only method you cannot use to solve x2 + 7x + 6 = 0 is using square roots. Sometimes one method is better for solving certain types of equations. The following table gives some advantages and disadvantages of the different methods.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Quadratic Formula
Common Errors
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Common Quadratic Formula Errors7x2 – 20x + 9
a = 7 b = -20 c = 9
Error #1: Entering b2 into the calculator
-202 = -400instead of +400
Correction:You should enter (-20)2 = 400
or just 202
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Common Quadratic Formula Errors
7x2 – 20x + 9
a = 7 b = -20 c = 9
Error #2:
Forgetting to take the square
root of the discriminate.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Common Quadratic Formula Errors
7x2 – 20x + 9
a = 7 b = -20 c = 9
Error #3: Entering –b for negative values of b.
- (-20) = +20
Holt Algebra 1
9-9 The Quadratic Formula and the DiscriminantCommon Quadratic Formula Errors
7x2 – 20x + 9
a = 7 b = -20 c = 9
Error #4: Forgetting that your calculator knows orders of operation and does exactly what you tell it to do.
6+4/2 = 8 Correct Answer = 5
Correction:
You should enter (6+4)/2 or
6+4, ENTER, Ans/2
2
46
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
L9-9 NOTES:
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
ASSIGNMENT:
• L9-9 pg 657 #9-69x3, skip #21, 54,
add #23, 53
#63 use another method
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
WARMUP
1. Solve –3x2 + 5x = 1 by using the Quadratic Formula.
2. Find the number of solutions of 5x2 – 10x – 8 = 0 by using the discriminant.
3. The height h in feet of an object shot straight up is modeled by h = –16t2 + vt + c, where c is the beginning height of the object above the ground. An object is shot up from 4 feet off the ground with an initial velocity of 48 feet per second. Will it reach a height of 40 feet? Use the discriminant to explain your answer.
4. Solve 8x2 – 13x – 6 = 0. Show your work.
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Lesson Quiz: Part I 1. Solve –3x2 + 5x = 1 by using the Quadratic
Formula.
2. Find the number of solutions of 5x2 – 10x – 8 = 0 by using the discriminant.
≈ 0.23, ≈ 1.43
2
Holt Algebra 1
9-9 The Quadratic Formula and the Discriminant
Lesson Quiz: Part II
The discriminant is zero. The object will reach its maximum height of 40 feet once.
4. Solve 8x2 – 13x – 6 = 0. Show your work.
3. The height h in feet of an object shot straight up is modeled by h = –16t2 + vt + c, where c is the beginning height of the object above the ground. An object is shot up from 4 feet off the ground with an initial velocity of 48 feet per second. Will it reach a height of 40 feet? Use the discriminant to explain your answer.