Solving Quadratic Equations Algebraically Lesson 2.2

Quadratic Equation A quadratic, or second degree, equation is one that can be written in the form: A quadratic, or second degree, equation is one that can be written in the form: ax 2 + bx + c = 0, for real constants a, b, and c, with a not equaling 0. Some strategies for solving equations Some strategies for solving equations Factoring Factoring Square rooting both sides Square rooting both sides Completing the square Completing the square Quadratic formula Quadratic formula

Zero Product Property If a product of real numbers is zero, then at least one of the factors is zero. In other words… if ab = 0, then a=0 or b=0 If a product of real numbers is zero, then at least one of the factors is zero. In other words… if ab = 0, then a=0 or b=0 Lets factor 3x 2 – x – 10 = 0, & 2x x = 6 Lets factor 3x 2 – x – 10 = 0, & 2x x = 6

x 2 = k If k < 0 there is 0 solutions If k < 0 there is 0 solutions If k = 0, then there is 1 solution named 0 If k = 0, then there is 1 solution named 0 If k > 0, then there is 2 solutions If k > 0, then there is 2 solutions √k and -√k √k and -√k 5(x – 3) 2 = 15 5(x – 3) 2 = 15

Completing the Square To complete the square of the expression x 2 + bx, add the square of one-half the coefficient of x, namely (b/x) 2. The addition produces a perfect square trinomial. To complete the square of the expression x 2 + bx, add the square of one-half the coefficient of x, namely (b/x) 2. The addition produces a perfect square trinomial. Lets try it with: 8x 2 – 24x + 3 = 0 Lets try it with: 8x 2 – 24x + 3 = 0 Page have lovely displays of completing the square Page have lovely displays of completing the square

Quadratic Formula

Your Turn Solve  4x 2 = -x – 2 Solve  4x 2 = -x – 2 The discriminant … The discriminant … Discriminant Value Number of Real Solutions b 2 – 4ac > 0 2 distinct Real Solutions b 2 – 4ac = 0 1 distinct real solution b 2 – 4ac < 0 0 real solutions

Polynomial Equations A polynomial equation of degree n is an equation that can be written in the form: A polynomial equation of degree n is an equation that can be written in the form: Where a n, …, a 0 are real numbers Page 94, 4x 4 – 13x = 0 f(x) = a n x n + a n-1 x n a 1 x + a 0 = 0