2. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Quadratic Equations 9
9.1
Introduction to Quadratic Equations
9.2
Solving Quadratic Equations by Completing the Square
9.3
The Quadratic Formula
9.4
Formulas
9.5
Applications and Problem Solving
9.6
Graphs of Quadratic Equations
9.7
Functions
(continued)
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3. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square a
Solve quadratic equations of the type ax2 = p. b
Solve quadratic equations of the type (x + c)2 = d c
Solve quadratic equations by completing the square. d
Solve certain applied problems involving quadratic equations of the type ax2 = p. b
Solve a system of two equations in two variables using the elimination method when multiplication is necessary.
(continued)
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4. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square
The Principle of Square Roots
The equation x2 = d has two real solutions when d > 0. The solutions are
The equation x2 = d has no real-number solution when d < 0.
The equation x2 = 0 has 0 as its only solution. a
Solve quadratic equations of the type ax2 = p.
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5. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square a
Solve quadratic equations of the type ax2 = p. A
Solve: x2 = 25
Solution We use the principle of square roots:
x2 = 25
x = or x = –5
We check mentally that 52 = 25 and (–5)2 = 25. The solutions are 5 and –5.
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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6. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square a
Solve quadratic equations of the type ax2 = p. B
Solve: a) x2 = 19 b) 9x2 = 27 c) 5x2  60 = 0
Solution a)
Check: x2 = x2 = 19
19 = = 19
The solutions are
(continued)
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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7. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square a
Solve quadratic equations of the type ax2 = p. A
Solve: a) x2 = 19 b) 9x2 = 27 c) 5x2  60 = 0
Solution
b) c)
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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8. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square b
Solve quadratic equations of the type (x + c)2 = d C
Solve: a) (x – 3)2 = 16 b) (x + 3)2 = 5
Solution a)
x – 3 = 4 or x – 3 = –4
x = 7 or x = –1
The solutions are 7 and –1. We leave the check to the student.
(continued)
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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9. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square b
Solve quadratic equations of the type (x + c)2 = d C
Solve: a) (x – 3)2 = 16 b) (x + 3)2 = 5
b) (x + 3)2 = 5
The solutions check and can be written as
(read as “negative three plus or minus the square root of five”).
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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10. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square b
Solve quadratic equations of the type (x + c)2 = d D
Solve: x2 + 8x + 16 = 17
Solution
x2 + 8x + 16 = 17
(x + 4)2 = 17
Sometimes we can factor an equation to express it as a square of a binomial.
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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11. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square
Completing the Square
To complete the square for an expression like x2 + bx, we take half of the coefficient of x and square it. Then we add that number, which is (b/2)2. c
Solve quadratic equations by completing the square.
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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12. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square c
Solve quadratic equations by completing the square. E
Complete the square: x2 – 16x
Solution
The coefficient of x is –16.
Half of –16 is –8 and (–8)2 = 64
Thus, x2 – 16 becomes a perfect-square trinomial when 64 is added:
x2 – 16x + 64 is the square of x – 8
The number 64 completes the square.
Check: (x – 8)2 = (x – 8)(x – 8) = x2 – 8x – 8x + 64
= x2 – 16x + 64
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13. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square c
Solve quadratic equations by completing the square. F
Solve by completing the square.
x2  8x = 15
Solution
Take half of 8 and square it to get 16.
We add 16 to both sides of the equation.
x2  8x + 16 = 
(x - 4) (x - 4) = 1
(x  4)2 = 1
x  4 = 1 or x  4 =  1
x = 5 or x = 3
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14. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square c
Solve quadratic equations by completing the square. G
Solve by completing the square.
x2  14x 7 = 0
Solution
We have x2  14x 7 = 0
x2  14x = 7
x2  14x + 49 =
(x  7)2 = 56
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15. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square c
Solve quadratic equations by completing the square. H
Solve by completing the square.
3x2 + 7x + 1 = 0
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16. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square c
Solve quadratic equations by completing the square. H
Solve by completing the square.
Solution
The coefficient of the x2 term must be 1. When it is not, we must multiply or divide on both sides to find an equivalent equation with an x2 coefficient of 1.
(continued)
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17. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square c
Solve quadratic equations by completing the square. H
Solve by completing the square.
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18. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square
Solving by Completing the Square
To solve a quadratic equation ax2 + bx + c = 0 by completing the square:
1. If a ≠ 1, multiply by 1/a so that the x2 –coefficient is 1.
2. If the x2 –coefficient is 1, add so that the equation is in the form x2 + bx = –c, or if step (1) has been applied.
3. Take half of the x-coefficient and square it. Add the result on both sides of the equation.
4. Express the side with the variables as the square of a binomial. c
Solve quadratic equations by completing the square.
(continued)
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19. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square
Solving by Completing the Square
To solve a quadratic equation ax2 + bx + c = 0 by completing the square:
5. Use the principle of square roots and complete the solution. c
Solve quadratic equations by completing the square.
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20. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square d
Solve certain applied problems involving quadratic equations of the type ax2 = p. I
The Taipei 101 tower in Taiwan is 1670 feet tall. How long would it take an object to fall to the ground from the top?
1. Familiarize. A formula that fits this situation is s = 16t2, where s is the distance, in feet, traveled by a body falling freely from rest in t seconds. We know that s is 1670 feet.
2. Translate. We know the distance is 1670 feet and that we need to solve for t.
1670 = 16t2
(continued)
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21. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square d
Solve certain applied problems involving quadratic equations of the type ax2 = p. I
3. Solve = 16t2
(continued)
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22. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
9.2
Solving Quadratic Equations by Completing the Square d
Solve certain applied problems involving quadratic equations of the type ax2 = p. I
4. Check. The number –10.2 cannot be a solution because time cannot be negative.
s = 16(10.2)2 = 16(104.04) =
This answer is close.
5. State. It takes about 10.2 seconds for an object to fall to the ground from the top of Taipei 101 .
Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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