1. Solving Quadratic Equations by the Quadratic Formula

2. The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation.

3. A quadratic equation written in standard form, ax 2 + bx + c = 0, has the solutions. The Quadratic Formula

4. Solve 11n 2 – 9n = 1 by the quadratic formula. 11n 2 – 9n – 1 = 0, so a = 11, b = -9, c = -1 The Quadratic Formula Example

5. x 2 + 8x – 20 = 0 (multiply both sides by 8) a = 1, b = 8, c =  20 Solve x 2 + x – = 0 by the quadratic formula. The Quadratic Formula Example

6. Solve x(x + 6) =  30 by the quadratic formula. x 2 + 6x + 30 = 0 a = 1, b = 6, c = 30 So there is no real solution. The Quadratic Formula Example

7. The expression under the radical sign in the formula (b 2 – 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. The Discriminant

8. Use the discriminant to determine the number and type of solutions for the following equation. 5 – 4x + 12x 2 = 0 a = 12, b = –4, and c = 5 b 2 – 4ac = (–4) 2 – 4(12)(5) = 16 – 240 = –224 There are no real solutions. The Discriminant Example

9. Solving Quadratic Equations Steps in Solving Quadratic Equations 1)If the equation is in the form (ax+b) 2 = c, use the square root property to solve. 2)If not solved in step 1, write the equation in standard form. 3)Try to solve by factoring. 4)If you haven’t solved it yet, use the quadratic formula.

10. Solve 12x = 4x 2 + 4. 0 = 4x 2 – 12x + 4 0 = 4(x 2 – 3x + 1) Let a = 1, b = -3, c = 1 Solving Equations Example

11. Solve the following quadratic equation. Solving Equations Example