X-Intercepts/Roots: Discriminant and the Quadratic Formula 1. Review: X-Intercepts are the Roots or Solutions x y Y = f(x) = 0 at the x-intercepts (curve crosses x-axis) The x-intercepts (where y=0) are called solutions, or the roots, or the zeroes of the quadratic function X-intercept/Root

X-Intercepts/Roots: Discriminant and the Quadratic Formula 2. How many Roots or X- Intercepts or Solutions do we have? Sometimes, we need to know how many X-Intercepts or Roots a quadratic function has How many times does it cross the x-axis? Does it even cross the x-axis at all? The value of the discriminant lets us know whether we have 0, 1 or 2 x-intercepts, roots or solutions The formula for the discriminant is: b 2 – 4acwhere a ≠ 0 (Remember: ax 2 + bx + c = 0)

X-Intercepts/Roots: Discriminant and the Quadratic Formula Value of Discriminant Number of Roots/Solutions Real or Imaginary Graph of Parabola (b 2 – 4ac) > 0 Positive Two (crosses x-axis in two places) Real (b 2 – 4ac) = 0 One (vertex is on x-axis) Real (b 2 – 4ac) < 0 Negative None (doesn’t cross x-axis) No Real, Only Imaginary 2. How many Roots or X- Intercepts or Solutions do we have?

X-Intercepts/Roots: Discriminant and the Quadratic Formula f(x) = 6x 2 – 2x – 3 a= 6, b= -2, c= -3 Ex 1 (I DO): Find the value of the discriminant then state the number of real or imaginary solutions. Discriminant: b 2 – 4ac = (-2) 2 – 4(6)(-3) = = 76 How many real or imaginary solutions? b 2 – 4ac > 0 means two real solutions 2. How many Roots or X- Intercepts or Solutions do we have? Discriminant: 76

X-Intercepts/Roots: Discriminant and the Quadratic Formula f(x) = -2x 2 – x – 1 = 0 a= -2, b= -1, c= -1 Ex 2 (WE DO): Find the value of the discriminant then state the number of real or imaginary solutions. Discriminant: b 2 – 4ac = (-1) 2 – 4(-2)(-1) = 1 – 8 = -7 How many real or imaginary solutions? b 2 – 4ac < 0 means no real solutions (only imaginary) 2. How many Roots or X- Intercepts or Solutions do we have? Discriminant: -7

X-Intercepts/Roots: Discriminant and the Quadratic Formula f(x) = 9x 2 – 3x – 8 = x 2 – 3x + 2 = 0 a= 9, b= -3, c= 2 Ex 3 (WE DO): Find the value of the discriminant then state the number of real or imaginary solutions. b 2 – 4ac = (-3) 2 – 4(9)(2) = 9 – 72 = -63 How many real or imaginary solutions? b 2 – 4ac < 0 means no real solutions (only imaginary) 2. How many Roots or X- Intercepts or Solutions do we have? Discriminant: -63

X-Intercepts/Roots: Discriminant and the Quadratic Formula 3. What is the Quadratic Formula?

X-Intercepts/Roots: Discriminant and the Quadratic Formula Ex 1 (I DO): What are the x-intercepts of the quadratic function? f(x) = x 2 + 4x – 5 a= 1, b= 4, c= What is the Quadratic Formula? and Discriminant is positive. 2 real solutions

X-Intercepts/Roots: Discriminant and the Quadratic Formula Ex 2 (WE DO): What are the x-intercepts of the quadratic function? f(x) = x 2 – 6x + 2 a= 1, b= -6, c= 2 3. What is the Quadratic Formula? and Discriminant is positive. 2 real solutions

X-Intercepts/Roots: Discriminant and the Quadratic Formula Ex 3 (WE DO): What are the x-intercepts of the quadratic function? f(x) = x 2 + 4x + 7 a= 1, b= 4, c= 7 3. What is the Quadratic Formula? Discriminant is negative. No real solutions (can’t take square root of negative number)

X-Intercepts/Roots: Discriminant and the Quadratic Formula Ex 4 (WE DO): What are the x-intercepts of the quadratic function? f(x) = x 2 + 8x + 16 a= 1, b= 8, c= What is the Quadratic Formula? Discriminant is 0. One real solutions