P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.
-
Upload
charity-gardner -
Category
Documents
-
view
228 -
download
0
Transcript of P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.
![Page 1: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/1.jpg)
P.1 QUADRATIC EQUATIONS AND APPLICATIONS
Copyright © Cengage Learning. All rights reserved.
![Page 2: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/2.jpg)
2
• Solve quadratic equations by factoring.
• Solve quadratic equations by extracting square roots.
• Solve quadratic equations by completing the square.
• Use the Quadratic Formula to solve quadratic equations.
• Use quadratic equations to model and solve real-life problems.
What You Should Learn
![Page 3: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/3.jpg)
3
Factoring
![Page 4: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/4.jpg)
4
Factoring
A quadratic equation in x is an equation that can be written in the general form
ax2 + bx + c = 0
where a, b, and c are real numbers with a 0. A quadratic equation in x is also called a second-degree polynomial equation in x.
In this section, you will study four methods for solving quadratic equations: factoring, extracting square roots, completing the square, and the Quadratic Formula.
![Page 5: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/5.jpg)
5
Factoring
The first method is based on the Zero-Factor Property.
If ab = 0, then a = 0 or b = 0.
To use this property, write the left side of the general form of a quadratic equation as the product of two linear factors.
Then find the solutions of the quadratic equation by setting each linear factor equal to zero.
Zero-Factor Property
![Page 6: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/6.jpg)
6
Example 1(a) – Solving a Quadratic Equation by Factoring
2x2 + 9x + 7 = 3
2x2 + 9x + 4 = 0
(2x + 1)(x + 4) = 0
2x + 1 = 0 x =
x + 4 = 0 x = –4
The solutions are x = and x = –4. Check these in the original equation.
Original equation
Write in general form.
Factor.
Set 1st factor equal to 0.
Set 2nd factor equal to 0.
![Page 7: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/7.jpg)
7
Example 1(b) – Solving a Quadratic Equation by Factoring
6x2 – 3x = 0
3x(2x – 1) = 0
3x = 0 x = 0
2x – 1 = 0 x =
The solutions are x = 0 and x = . Check these in the original equation.
Original equation
Set 1st factor equal to 0.
Factor.
Set 2nd factor equal to 0.
cont’d
![Page 8: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/8.jpg)
8
Factoring
Be sure you see that the Zero-Factor Property works only for equations written in general form (in which the right side of the equation is zero).
So, all terms must be collected on one side before factoring.
For instance, in the equation (x – 5)(x + 2) = 8, it is incorrect to set each factor equal to 8.
To solve this equation, you must multiply the binomials on the left side of the equation, and then subtract 8 from each side.
![Page 9: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/9.jpg)
9
Factoring
After simplifying the left side of the equation, you can use the Zero-Factor Property to solve the equation.
Try to solve this equation correctly.
![Page 10: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/10.jpg)
10
Extracting Square Roots
![Page 11: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/11.jpg)
11
Extracting Square Roots
Solving an equation of the form u2 = d without going through the steps of factoring is called extracting square roots.
![Page 12: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/12.jpg)
12
Example 2 – Extracting Square RootsSolve each equation by extracting square roots.
a. 4x2 = 12 b. (x – 3)2 = 7
Solution:
a. 4x2 = 12
x2 = 3
x =
When you take the square root of a variable expression,
you must account for both positive and negative solutions.
Write original equation.
Divide each side by 4.
Extract square roots.
![Page 13: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/13.jpg)
13
Example 2 – Solution
So, the solutions are x = and x = – . Check these in the original equation.
b. (x – 3)2 = 7
x – 3 =
x = 3
The solutions are x = 3 . Check these in the original equation.
Write original equation.
Extract square roots.
Add 3 to each side.
cont’d
![Page 14: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/14.jpg)
14
Completing the Square
![Page 15: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/15.jpg)
15
Completing the Square
Note that when you complete the square to solve a quadratic equation, you are just rewriting the equation so it can be solved by extracting square roots.
![Page 16: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/16.jpg)
16
Example 3 – Completing the Square: Leading Coefficient Is 1
Solve x2 + 2x – 6 = 0 by completing the square.
Solution:
x2 + 2x – 6 = 0
x2 + 2x = 6
x2 + 2x + 12 = 6 + 12
Write original equation.
Add 6 to each side.
Add 12 to each side.
![Page 17: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/17.jpg)
17
Example 3 – Solution
(x + 1)2 = 7
x + 1 =
x = –1
The solutions are x = –1 . Check these in the original equation as follows.
Simplify.
Take square root of each side.
Subtract 1 from each side.
cont’d
![Page 18: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/18.jpg)
18
Example 3 – Solution
Check:
x2 + 2x – 6 = 0
(–1 + )2 + 2 (–1 + ) – 6 ≟ 0
8 – 2 – 2 + 2 – 6 ≟ 0
8 – 2 – 6 = 0
Check the second solution in the original equation.
Write original equation.
Substitute –1 + for x.
Multiply.
Solution checks.
cont’d
![Page 19: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/19.jpg)
19
Completing the Square
When solving quadratic equations by completing the square, you must add (b/2)2 to each side in order to maintain equality.
If the leading coefficient is not 1, you must divide each side of the equation by the leading coefficient before completing the square.
![Page 20: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/20.jpg)
20
The Quadratic Formula
![Page 21: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/21.jpg)
21
The Quadratic Formula
The Quadratic Formula is one of the most important formulas in algebra. You should learn the verbal statement of the Quadratic Formula:
“Negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.”
![Page 22: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/22.jpg)
22
The Quadratic Formula
In the Quadratic Formula, the quantity under the radical sign, b2 – 4ac, is called the discriminant of the quadratic expression ax2 + bx + c. It can be used to determine the nature of the solutions of a quadratic equation.
![Page 23: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/23.jpg)
23
The Quadratic Formula
If the discriminant of a quadratic equation is negative, as in case 3 above, then its square root is imaginary (not a real number) and the Quadratic Formula yields two complex solutions.
When using the Quadratic Formula, remember that before the formula can be applied, you must first write the quadratic equation in general form.
![Page 24: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/24.jpg)
24
Example 6 – The Quadratic Formula: Two Distinct Solutions
Use the Quadratic Formula to solve x2 + 3x = 9.
Solution:
The general form of the equation is x2 + 3x – 9 = 0. The discriminant is b2 – 4ac = 9 + 36 = 45, which is positive. So, the equation has two real solutions.
You can solve the equation as follows.
x2 + 3x – 9 = 0 Write in general form.
Quadratic Formula
![Page 25: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/25.jpg)
25
Example 6 – Solution
The two solutions are:
Check these in the original equation.
Substitute a = 1, b = 3,and c = –9.
Simplify.
Simplify.
cont’d
![Page 26: P.1 QUADRATIC EQUATIONS AND APPLICATIONS Copyright © Cengage Learning. All rights reserved.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649de75503460f94ae1609/html5/thumbnails/26.jpg)
26
The Quadratic Formula: One Solution
Use the Quadratic Formula to solve 8x2 – 24x + 18 = 0.