Section 9.2 Using the Square Root Property and Completing the Square to Find Solutions

Square Root Property Square Root Property If x2 = a, then x = or x = for all nonnegative real numbers a. 2

Example Solve each equation for x. 3

Example Solve. 3x2 + 5x = 18 + 5x + x2 Simplify the equation by placing all the variable terms on the left and the constants on the right. 3x2 + 5x = 18 + 5x + x2 2x2 = 18 x2 = 9 x2 = ±3 4

Example Solve. Take the square root of each side. Simplify. Subtract 3 from both sides. Divide both sides by 2. 5

Or the roots can be written Example Solve. We need a +36 to complete the square. Add +36 to both sides of the equation. Subtract 3 from both sides. Divide both sides by 2. Or the roots can be written 6

Completing the Square If a quadratic equation is not in a form where the square root property can be used, the equation can be rewritten. This is called completing the square. Completing the Square Put the equation in the form ax2 + bx = c. If a  1, divide each term by a. Square and add the result to both sides of the equation. Factor the left side (a perfect square trinomial). Use the square root property. Solve the equations. Check the solutions in the original equation. 7

Example Solve. 2x2 + 5x = 3 Step 1: 2x2 + 5x = 3 Divide each term by 2. Take one-half the coefficient of x and square it. Step 2: Add to each side. Factor the left side. Continued 8

Example (cont) Solve. 2x2 + 5x = 3 Step 4: Step 5: Use the square root property. Simplify. Solve for x. Step 5: Continued 9

Example (cont) Solve by completing the square. 2x2 + 5x = 3 Step 6:  ? Step 6: ?  Both values check. ? ?  10