10 Quadratic Equations

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 1. 2. Solve quadratic equations by completing the square when the coefficient of the second-degree term is not 1. 3. Simplify the terms of an equation before solving. 4. Solve applied problems that require quadratic equations.

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.

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

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

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.

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.

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.

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

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.

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

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 = –16t2 + 128t, where s is in feet. At what times will the ball be 48 feet above the ground? –16t2 + 128t = 48 t2 – 8t + 16 = –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.