Objectives: To utilize the Quadratic Formula. To describe the nature of the roots and decide if a quardratic is factorable, using the discriminant. Adapted from resources at:

Quadratic Formula If you take a quadratic equation in standard form (ax 2 +bx+c=0), and you complete the square, you will get the quadratic formula!

X equals negative b  X equals negative b Plus or minus root  Plus or minus root b squared minus 4ac  b squared minus 4ac All over 2a  All over 2a image from:

When Should I Use the Quadratic Formula? When: the equation is not factorable or when you’re stuck on how to factor the equation or there’s a leading coefficient, so that completing the square would be cumbersome

The discriminant tells you how many solutions and what type you will have. If the discriminant is:  positive –> 2 real solutions a perfect square -> 2 rational solutions  is factorable  zero –>1 real, rational solution  is factorable  negative –> 2 complex solutions Discriminant: b 2 -4ac It doesn’t hit the x-axis.

Examples Find the discriminant and give the number and type of solutions. a. 9x 2 +6x+1=0 a=9, b=6, c=1 b 2 -4ac=(6) 2 -4(9)(1) =36-36=0 1 real, rational solutionFactorable! b. 9x 2 +6x=4 9x 2 +6x-4=0 a=9, b=6, c=-4 b 2 -4ac=(6) 2 -4(9)(-4) =36+144=180 2 real, irrational solutions c. 9x 2 +6x+5=0 a=9, b=6, c=5 b 2 -4ac=(6) 2 -4(9)(5) =36-180= complex solutions

Examples 1. 3x 2 +8x=35 3x 2 +8x-35=0 a=3, b=8, c= -35 OR

2. -2x 2 =-2x+3 -2x 2 +2x-3=0 a=-2, b=2, c= -3