Transcript Chapter 9 Section 8

9-8 The Quadratic Formula
Preview
Warm Up
California Standards
Lesson Presentation
9-8 The Quadratic Formula
Warm Up
Evaluate for x = –2, y = 3, and z = –1.
1. x2
4
2. xyz 6
3. x2 – yz 7
4. y – xz 1
5. –x 2
6. z2 – xy 7
9-8 The Quadratic Formula
California
Standards
19.0 Students know the quadratic
formula and are familiar with its proof by
completing the square.
20.0 Students use the quadratic
formula to find the roots of a seconddegree polynomial and to solve quadratic
equations.
9-8 The Quadratic Formula
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.
9-8 The Quadratic Formula
9-8 The Quadratic Formula
9-8 The Quadratic Formula
9-8 The Quadratic Formula
Remember!
To add fractions, you need a common
denominator.
9-8 The Quadratic Formula
Additional Example 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.
Substitute 6 for a, 5 for b,
and –4 for c.
Simplify.
9-8 The Quadratic Formula
Additional Example 1A Continued
Solve using the Quadratic Formula.
6x2 + 5x – 4 = 0
Simplify.
Write as two equations.
Solve each equation.
9-8 The Quadratic Formula
Additional Example 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.
Substitute 1 for a, –1 for b,
and –20 for c.
Simplify.
9-8 The Quadratic Formula
Additional Example 1B Continued
Solve using the Quadratic Formula.
x2 = x + 20
Simplify.
Write as two equations.
x=5
or
x = –4
Solve each equation.
9-8 The Quadratic Formula
Helpful Hint
You can graph the related quadratic function to
see if your solutions are reasonable.
9-8 The Quadratic Formula
Check It Out! Example 1a
Solve using the Quadratic Formula. Check your
answer.
–3x2 + 5x + 2 = 0
–3x2 + 5x + 2 = 0
Identify a, b, and c.
Use the Quadratic Formula.
Substitute –3 for a, 5 for b,
and 2 for c.
Simplify.
9-8 The Quadratic Formula
Check It Out! Example 1a Continued
Solve using the Quadratic Formula. Check your
answer.
–3x2 + 5x + 2 = 0
Simplify.
Write as two equations.
x=–
or
x=2
Solve each equation.
9-8 The Quadratic Formula
Check It Out! Example 1a Continued
Solve using the Quadratic Formula. Check your
answer.
Check –3x2 + 5x + 2 = 0
x=–
or
x=2
9-8 The Quadratic Formula
Check It Out! Example 1b
Solve using the Quadratic Formula. Check your
answer.
2 – 5x2 = –9x
(–5)x2 + 9x + (2) = 0
Write in standard form. Identify
a, b, and c.
Use the Quadratic Formula.
Substitute –5 for a, 9 for b,
and 2 for c.
Simplify
9-8 The Quadratic Formula
Check It Out! Example 1b Continued
Solve using the Quadratic Formula. Check your
answer.
2 – 5x2 = –9x
Simplify.
Write as two equations.
x=–
or x = 2
Solve each equation.
9-8 The Quadratic Formula
Check It Out! Example 1b Continued
Solve using the Quadratic Formula. Check your
answer.
Check –5x2 + 9x + 2 = 0
–5(2)2 + 9(2) + 2
–20 + 18 + 2
0
0
0
0
–5x2 + 9x + 2 = 0
–5
+9
+2
0
+2
0
0
0
9-8 The Quadratic Formula
Because the Quadratic Formula contains a square
root, the solutions may be irrational. You can give
the exact solution by leaving the square root in
your answer, or you can approximate the solutions.
9-8 The Quadratic Formula
Additional Example 2: Using the Quadratic Formula
to Estimate Solutions
Solve x2 + 3x – 7 = 0 using the Quadratic Formula.
Check reasonableness
Estimate
: x ≈ 1.54 or x ≈ –4.54.
9-8 The Quadratic Formula
Check It Out! Example 2
Solve 2x2 – 8x + 1 = 0 using the Quadratic Formula.
Check reasonableness
Estimate
: x ≈ 3.87 or x ≈ 0.13.
9-8 The Quadratic Formula
There is no one correct way to solve a quadratic
equation. Many quadratic equations can be
solved using several different methods:
graphing, factoring, completing the square,
using square roots, and using the Quadratic
Formula.
9-8 The Quadratic Formula
Additional Example 3: Solving Using Different Methods
Solve x2 – 9x + 20 = 0. Show your work. Use
at least two different methods. Check your
answer.
Method 1 Solve by graphing.
Write the related quadratic
2
y = x – 9x + 20
function and graph it.
The solutions are the xintercepts, 4 and 5.
9-8 The Quadratic Formula
Additional Example 3 Continued
Solve x2 – 9x + 20 = 0. Show your work. Use
at least two different methods. Check your
answer.
Method 2 Solve by factoring.
x2 – 9x + 20 = 0
(x – 4)(x – 5) = 0
Factor.
x – 4 = 0 or x – 5 = 0
Use the Zero Product Property.
x = 4 or x = 5
Solve each equation.
9-8 The Quadratic Formula
Additional Example 3 Continued
Solve x2 – 9x + 20 = 0. Show your work. Use
at least two different methods. Check your
answer.
Check: 4 and 5
Check
x2 – 9x + 20 = 0
(4)2 – 9(4) + 20 0
16 – 36 + 20 0
0 0
x2 – 9x + 20 = 0
(5)2 – 9(5) + 20
25 – 45 + 20
0
0
0
0
9-8 The Quadratic Formula
Check It Out! Example 3a
Solve. Show your work and check your answer.
x2 + 7x + 10 = 0
Method 3 Solve by completing the square.
x2 + 7x + 10 = 0
x2 + 7x = –10
x2 +7x
= –10
Add
to both sides.
Factor and simplify.
Take the square root of
both sides.
9-8 The Quadratic Formula
Check It Out! Example 3a Continued
Solve. Show your work and check your answer.
x2 + 7x + 10 = 0
or
Solve each equation.
x = –2 or x = –5
Check x2 + 7x + 10 = 0
(–2)2 + 7(–2) + 10
4 – 14 + 10
0
0
0
0
x2 + 7x + 10 = 0
(–5)2 + 7(–5) + 10
25 – 35 + 10
0
0
0
0
9-8 The Quadratic Formula
Check It Out! Example 3b
Solve. Show your work and check your answer.
–14 + x2 – 5x = 0
Method 4 Solve using the Quadratic Formula.
x2 – 5x – 14 = 0
1x2 – 5x – 14 = 0
Identify a, b, and c.
Substitute 1 for a, –5 for b,
and –14 for c.
Simplify.
9-8 The Quadratic Formula
Check It Out! Example 3b Continued
Solve. Show your work and check your answer.
–14 + x2 – 5x = 0
x=7
or
Write as two equations.
or x = –2
Solve each equation.
9-8 The Quadratic Formula
Check It Out! Example 3b Continued
Solve. Show your work and check your answer.
–14 + x2 – 5x = 0
Check x2 – 5x – 14 = 0
x2 – 5x – 14 = 0
72 – 5(7) – 14
0
–22 – 5(–2) – 14
0
49 – 35 – 14
14 – 14
0
0
4 + 10 – 14
14 – 14
0
0
0
0
0
0
9-8 The Quadratic Formula
Check It Out! Example 3c
Solve. Show your work and check your answer.
2x2 + 4x – 21 = 0
Method 1 Solve by graphing.
Write the related quadratic
2x2 + 4x – 21 = y
function.
Divide each term by 2 and
graph.
The solutions are the
x-intercepts and appear to
be ≈ 2.4 and ≈ –4.4.
9-8 The Quadratic Formula
Check It Out! Example 3c Continued
Solve. Show your work and check your answer.
2x2 + 4x – 21 = 0
Check
y = 2x2 + 4x – 21 using a calculator.
Answers x ≈ –4.39 and ≈ 2.39 appear to be
reasonable.
9-8 The Quadratic Formula
Notice that all of the methods in Example 3 on
p. 600 produce the same solution, –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 table below
gives some advantages and disadvantages of
the different methods.
9-8 The Quadratic Formula
9-8 The Quadratic Formula
Lesson Quiz
1. Solve x2 + x = 12 by using the Quadratic Formula.
3, –4
2. Solve –3x2 + 5x = 1 by using the Quadratic
Formula. = 0.23, ≈ 1.43
3. Solve 8x2 – 13x – 6 = 0. Use at least 2 different
methods.