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4.6 The Quadratic Formula and the Discriminant. Objectives: Solve quadratic equations by using the Quadratic Formula Use the discriminant to determine the number and type of roots for a quadratic equation. Quadratic Formula.
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4.6 The Quadratic Formula and the Discriminant Objectives: Solve quadratic equations by using the Quadratic Formula Use the discriminant to determine the number and type of roots for a quadratic equation.
Quadratic Formula Always works to solve a quadratic equation, but is a little lengthy. The solutions of a quadratic equation of the form where a≠0 are given by the formula:
Example x²-8x=33 x²-8x-33=0 Set = 0 a=1, b=-8, c=-33
Another Example 7x²+6x+2=0 a=7, b=6, c=2 Since ALL of the coefficients are divisible by 2, simplify by dividing them by 2.
Discriminant The discriminant describes the solution to a quadratic equation. The part of the quadratic formula under the radical is the discriminant or b²-4ac. • If b2 – 4ac > 0, and a perfect square • You have two rational roots • If b2 – 4ac >0, and not a perfect square. • You have two irrational roots • If b2 – 4ac = 0 • You have 1 real, rational root. (Repeated root) • If b2 – 4ac < 0 • You have two complex roots
Examples Find the discriminant and describe the number and type of roots. • x²-16x+64=0 b²-4ac (-16)²-4(1)(64)= 256-256=0 One real, rational root because the discriminant equals 0. b. 7x²-3x=0 (-3)²-4(7)(0)= 9-0=9 Two rational roots because 9 is positive and a perfect square. • 3x²-x+5=0 (-1)²-4(3)(5)= 1-60=-59 Two complex roots because the discriminant is a negative.
We have discussed several methods for solving quadratic equations – which one do you use?
Solve – use any method. 1. 7x²-14x=0 7x(x-2)=0 x=0, x=2 • x²-64=0 x²=64 x=8 • x²-16x+64=0 (x-8)(x-8)=0 x=8 4. x²+5x+8=0 Doesn’t factor, not easily done by completing the square (5 is odd) so use quadratic formula.
