Quadratic Equations

Solving ax 2 + bx + c = 0 Taking Square Roots of Both Sides Graphing Factoring Completing the Square Quadratic Formula

Taking Square Root of Both Sides When your “b” term is zero you can set up a quadratic equation so that it is solvable by taking the square root of both sides. Example: x 2 – 81 = 0 Example: x = 0

Graphing Quadratics Vertex is (-b/2a, f(-b/2a)) Vertex form is y = a(x-h) 2 + k where (h,k) is the vertex Example: y = 2x 2 + 4x + 9

Graphing Quadratics Graph on calculator If there are real roots, you will be able to identify them as the x-intercepts and you can trace or use the table to estimate the roots If the roots are imaginary the quadratic will not show any x-intercepts and you will have to use another method to solve the quadratic Ex: y = x 2 – 2x + 5

Factoring Quadratics Look for any greatest common factors that can factor out first Look for any special factoring patterns Guess and check, use FOIL to check your work Set the factors equal to zero and use the Zero Product Property to find the solutions

Factoring Quadratics Examples: x 2 – 6x – 16 = 0 x 2 – 2x – 8 = 0 x 2 – 9 = 0 x 2 + 4x + 4 = 0 2 x 2 – 32 = 0 4 x x -5 = 0

Completing the Square The point of this procedure is to get a trinomial arranged so that it can be factored into a “perfect square trinomial”. What is a “perfect square trinomial? Example: (x + 3) 2 Example: (x – 5) 2

Perfect Square Trinomials (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 - 2ab + b 2 Example: Find the number that would make the quadratic “perfect” X 2 + 6x + _______ X 2 + 8x + _______ X x + _______ X 2 + 5x + _______

Completing the Square X x -1 = 0 4X x + 1 = 0

The Quadratic Formula If you complete the square on the general quadratic equation, you get the quadratic formula.

X 2 – 8x + 15 = 0 3X 2 + 5x = 2 X = 8x