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6.4 Completing t he Square. Objectives: Solve quadratic equations by using the Square Root Property Solve quadratic equations by completing the square. Square Root Property. For any real number n, if x²=n, then For example:
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6.4 Completing the Square Objectives: Solve quadratic equations by using the Square Root Property Solve quadratic equations by completing the square.
Square Root Property For any real number n, if x²=n, then For example: Sometimes you may have to factor the left side first. What happens when it won’t factor? You have to complete the square! (Force it to factor)
Perfect Square Trinomials Perfect square trinomials factor to be the product of the same binomial such as (x-3)(x-3) or (x-3)². The trinomial is written in the form a²+2ab+b² or a²-2ab+b². If you know the first two terms you can figure out what the last term would be by dividing the 2nd term by 2 and squaring it. What value for c would make the trinomial a perfect square? x²+12x+c 122=6 6²=36 so x²+12x+36 is a perfect square trinomial. It factors to be (x+6)²
Factor these perfect square trinomials. • x²-6x+9 • x²+14x+49 • 9x²-30x+25 • 16x²+8x+1
Completing the Square (always works to solve a quadratic equation!) • Get equation in the form x²+bx=c • Find half of b. • Square the result (half of b) • Add this to both sides. • Factor the left side, simplify the right side. • Take the square root of each side to solve. (Remember to use ) • Solve.
Example Solve by completing the square. x²+6x+8=0 x²+6x=-8 62=3 3²=9 add 9 to both sides x²+6x+9=-8+9 (x+3)²=1 x+3=1 x+3=1 x+3=-1 x=-2, x=-4 Solutions: {-4, -2}
What happens when a≠1 Divide all terms by the coefficient of x². Complete the square. Example: 3x²-2x-1=0 Divide everything by 3 x²-⅔x-⅓=0 Move -⅓ over x²-⅔x=⅓ Divide -⅔ by 2 and square -⅔2=-⅓ (-⅓)²= Add to both sides Factor left, add numbers on right Take the sq. root of both sides Solve
