Aim: The Discriminant Course: Adv. Alg, & Trig. Aim: What is the discriminant and how does it help us determine the roots of a parabola? Do Now: Graph x 2 – 2x – 3 = y x 2 – 6x + 7 = y x 2 – 4x + 4 = y x 2 – 4x + 5 = y Describe the roots for each.

Aim: The Discriminant Course: Adv. Alg, & Trig. The Graph, the Roots, & the x-axis y = ax 2 + bx + cEquation of parabola y = 0 2 real roots 2 real equal roots NO real roots, complex

Aim: The Discriminant Course: Adv. Alg, & Trig. Parabolas x 2 – 4x + 5 = y x 2 – 2x – 3 = yx 2 – 6x + 7 = y x 2 – 4x + 4 = y Imaginary roots 2 real rational roots {-1 and 3} 2 real rational roots that are equal {2} 2 real irrational roots

Aim: The Discriminant Course: Adv. Alg, & Trig. x 2 – 2x – 3 = 0 {-1 and 3} Quadratic Formula Solutions x 2 – 6x + 7 = 0x 2 – 4x + 4 = 0 {2} x 2 – 4x + 5 = 0

Aim: The Discriminant Course: Adv. Alg, & Trig. The Discriminant Knows! x 2 – 4x + 5 = y x 2 – 2x – 3 = yx 2 – 6x + 7 = y x 2 – 4x + 4 = y Imaginary roots 2 real rational roots {-1 and 3} 2 real rational roots that are equal {2} 2 real irrational roots

Aim: The Discriminant Course: Adv. Alg, & Trig. The Discriminant The discriminant - the expression under the radical sign. It determines the nature of the roots of a quadratic equation when a, b, and c are rational numbers. b 2 – 4ac Quadratic Formula

Aim: The Discriminant Course: Adv. Alg, & Trig. The Nature of the Roots - Case 1 x 2 – 2x – 3 = y 2 real rational roots {-1 and 3} b 2 – 4ac = If the b 2 – 4ac > 0 and b 2 – 4ac is a perfect square, then the roots of the equation ax 2 +bx + c = 0 are real, rational and unequal. (-2) 2 – 4(1)(-3) the discriminant = 16 the discriminant is a perfect square a = 1, b = -2, c = -3

Aim: The Discriminant Course: Adv. Alg, & Trig. The Nature of the Roots - Case 2 b 2 – 4ac = If the b 2 – 4ac > 0 and b 2 – 4ac is not a perfect square, then the roots of the equation ax 2 +bx + c = 0 are real, irrational and unequal. (-6) 2 – 4(1)(7) the discriminant 36 – 28 = 8 the discriminant is a positive number, but not a perfect squ. a = 1, b = -6, c = 7x 2 – 6x + 7 = y 2 real irrational roots

Aim: The Discriminant Course: Adv. Alg, & Trig. The Nature of the Roots - Case 3 b 2 – 4ac = If the b 2 – 4ac = 0, then the roots of the equation ax 2 +bx + c = 0 are real, rational and equal. (-4) 2 – 4(1)(4) the discriminant 16 – 16 = 0 the discriminant is zero a = 1, b = -4, c = 4x 2 – 4x + 4 = y 2 real rational roots that are equal {2}

Aim: The Discriminant Course: Adv. Alg, & Trig. The Nature of the Roots - Case 4 b 2 – 4ac = If the b 2 – 4ac < 0, then the roots of the equation ax 2 + bx + c = 0 are imaginary. (-4) 2 – 4(1)(5) the discriminant 16 – 10 = -4 the discriminant is a negative number a = 1, b = -4, c = 5x 2 – 4x + 5 = y Imaginary roots

Aim: The Discriminant Course: Adv. Alg, & Trig. The Discriminant Value of DiscriminantNature of roots of ax 2 + bx + c = 0 b 2 - 4ac > 0 and b 2 - 4ac is a perfect square real, rational, unequal b 2 - 4ac > 0 and b 2 - 4ac is not a perfect square real, irrational, unequal b 2 - 4ac = 0real, rational, equal b 2 - 4ac < 0imaginary

Aim: The Discriminant Course: Adv. Alg, & Trig. Model Problem The roots of a quadratic equation are real, rational, and equal when the discriminant is 1)-2 2)2 3)0 4)4 The roots of the equation 2x 2 – 4 = 4 are 1)real and irrational 2)real, rational and equal 3)real, rational and unequal 4)imaginary

Aim: The Discriminant Course: Adv. Alg, & Trig. Model Problem Find the largest integral value of k for which the roots of the equation 2x 2 + 7x + k = 0 are real. a = 2, b = 7, c = k If the roots are real, then the discriminant b 2 - 4ac > 0. substitute into b 2 - 4ac ≥ 0 (-7) 2 - 4(2)(k) ≥ (k) ≥ 0 49 ≥ 8k 6 1/8 ≥ k The largest integer: k = 6 = c = 49 – 56 = -7 check: c = = 49 – 48 = 1 c = 6 imaginary