1. 10
Quadratic Equations

2. 10.2 Solving Quadratic Equations by Completing the Square
Objectives
1. Solve quadratic equations by completing the square when the coefficient of the second-degree term is Solve quadratic equations by completing the square when the coefficient of the second-degree term is not Simplify the terms of an equation before solving. 4. Solve applied problems that require quadratic equations.

3. Complete the Square When the Leading Coefficient is 1
Recall from Section 6.5 that a perfect square trinomial has the form
x2 + 2kx + k2 or x2 – 2kx + k2,
where k represents a positive number.

4. Complete the Square When the Leading Coefficient is 1
Example 1 Complete each trinomial so it is a perfect square. Then factor the trinomial.
(a) x2 + 6x + ____
The perfect square trinomial will have the form x2 + 2kx + k2. Thus, the middle term, 6x, must equal 2kx.
The perfect square trinomial is

5. Complete the Square When the Leading Coefficient is 1
Example 1 (continued) Complete each trinomial so it is a perfect square. Then factor the trinomial.
(b) x2 – 5x + ____
The perfect square trinomial will have the form x2  2kx + k2. Thus, the middle term, – 5x, must equal 2kx.
The perfect square trinomial is

6. Complete the Square When the Leading Coefficient is 1
Example 2
Solve x2 + 8x + 9 = 0.
x2 + 8x = –9
Subtract 9 from each side.
x2 + 8x + 16 = –9 + 16
Add 16 to complete the square.
(x + 4)2 = 7
Factor, add.
Now use the square root property to complete the solution.

7. Complete the Square When the Leading Coefficient is Not 1
Solving a Quadratic Equation by Completing the Square
Step 1 Be sure the second-degree term has coefficient 1. If the coefficient of the second-degree term is 1, proceed to Step 2. If it is not 1, but some other non-zero number a, divide each side of the equation by a.
Step 2 Write in correct form. Make sure that all terms with variables are on one side of the equals sign and that all constant terms are on the other side.
Step 3 Complete the square. Take half the coefficient of the first-degree term, and square it. Add the square to each side of the equation. Factor the variable side, and simplify on the other side.
Step 4 Solve the equation by using the square root property.

8. Complete the Square When the Leading Coefficient is Not 1
Example 5
Solve 4x2 – 4x – 15 = 0.
Before completing the square, the coefficient of x2 must be 1.
We get 1 as the coefficient of x2 here by dividing each side by 4.
Divide by 4.

9. Complete the Square When the Leading Coefficient is Not 1
Example 4 (concluded)
Solve 4x2 – 4x – 15 = 0.
From previous slide.
Factor, add.
Square root property

10. Simplify the Terms of an Equation Before Solving
Example 7
Solve (x + 4)(x – 2) = –6.
x2 + 2x – 8 = –6
Multiply using the FOIL method.
x2 + 2x = 2
Add 8.
Add (½ · 2)2 = 1 to complete the square.
x2 + 2x + 1 = 2 + 1
(x + 1)2 = 3
Factor, add.
Square root property.

11. Solve Quadratic Equations with Only One Solution
Note
The solutions given in Example 7 are exact.
In applications, decimal solutions are more appropriate.
Using the square root key of a calculator,
Approximating the two solutions gives

12. Solve Applied Problems that Require Quadratic Equations
Example 8
Suppose a ball is thrown upward from ground level with an initial velocity of 128 feet per second. Its altitude (height) s at time t is given by s = –16t t, where s is in feet. At what times will the ball be 48 feet above the ground?
–16t t = 48
t2 – 8t = –3 + 16
(t – 4)2 = 13
16t2 – 128t = –48
t2 – 8t = –3
Thus, the is 48 feet high at two times: at 7.6 sec and 0.4 sec.