9-8 Completing the Square
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1 In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. X 2 + 6x + 9 x 2 8x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.
2 An expression in the form x 2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x 2 + bx to form a trinomial that is a perfect square. This is called completing the square.
3 Example 1: Completing the Square Complete the square to form a perfect square trinomial. A. x 2 + 2x + B. x 2 6x + x 2 + 2x Identify b.. x 2 + 6x x 2 + 2x + 1 x 2 6x + 9
4 Check It Out! Example 1 Complete the square to form a perfect square trinomial. a. x x + b. x 2 5x + x x Identify b..
5 To solve a quadratic equation in the form x 2 + bx = c, first complete the square of x 2 + bx. Then you can solve using square roots.
6 Solving a Quadratic Equation by Completing the Square
7 Example 2A: Solving x 2 +bx = c Solve by completing the square. x x = 15 Step 1 x x = 15 The equation is in the form x 2 + bx = c. Step 2 Step 3 x x + 64 = Step 4 (x + 8) 2 = 49 Step 5 x + 8 = 7 Step 6 x + 8 = 7 or x + 8 = 7 x = 1 or x = 15. Complete the square. Factor and simplify. Take the square root of both sides. Write and solve two equations.
8 Example 2A Continued Solve by completing the square. x x = 15 The solutions are 1 and 15. Check x x = 15 x x = 15 ( 1) ( 1) ( 15) ( 15)
9 Example 2B: Solving x 2 +bx = c Solve by completing the square. x 2 4x 6 = 0 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Write in the form x 2 + bx = c. Complete the square. Factor and simplify.. Take the square root of both sides. Write and solve two equations.
10 Example 3A: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. Step 1 3x x 15 = 0 x 2 4x + 5 = 0 x 2 4x = 5 x 2 + ( 4x) = 5 Divide by 3 to make a = 1. Write in the form x 2 + bx = c. Step 2 Step 3 x 2 4x + 4 = Complete the square.
11 Example 3A Continued Solve by completing the square. 3x x 15 = 0 Step 4 (x 2) 2 = 1 Factor and simplify. There is no real number whose square is negative, so there are no real solutions.
12 Example 3B: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. 5x x = 4 Step 1 Divide by 5 to make a = 1. Write in the form x 2 + bx = c. Step 2.
13 Example 3B Continued Solve by completing the square. Step 3 Complete the square. Rewrite using like denominators. Step 4 Factor and simplify. Step 5 Take the square root of both sides.
14 Example 3B Continued Solve by completing the square. Step 6 Write and solve two equations.
15 Example 4: Problem-Solving Application A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary. 1 Understand the Problem The answer will be the length and width of the room. List the important information: The room area is 195 square feet. The width is 2 feet less than the length.
16 2 Make a Plan Example 4 Continued Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation.
17 Example 4 Continued 3 Solve Let x be the width. Then x + 2 is the length. Use the formula for area of a rectangle. l w = A length times width = area of room x + 2 x = 195
18 Step 1 x 2 + 2x = 195 Example 4 Continued Simplify. Step 2. Step 3 x 2 + 2x + 1 = Complete the square by adding 1 to both sides. Step 4 (x + 1) 2 = 196 Factor the perfect-square trinomial. Step 5 x + 1 = 14 Take the square root of both sides. Step 6 x + 1 = 14 or x + 1 = 14 Write and solve two equations. x = 13 or x = 15
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