The Quadratic Formula What Does The Formula Do

The Quadratic Formula.

What Does The Formula Do ? The Quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise. The formula states that for a quadratic equation of the form : ax 2 + bx + c = 0 The roots of the quadratic equation are given by :

• The Quadratic Formula – The solutions of a quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, are given by the following formula:

• Solve Two Rational Roots by using the Quadratic Formula. a=1 b = -12 c = -28 Solutions are -2 and 14.

• Solve One Rational Root by using the Quadratic Formula. a=1 b = 22 c = 121 • The solution is .

• Solve Irrational Roots by using the Quadratic Formula. a=2 b=4 c = -5 • The exact solutions are and .

Complex Roots **SKIP SECTION** • Solve • The exact solutions are by using the Quadratic Formula. and .

Roots and the Discriminant • The Discriminant = b 2 – 4 ac • The value of the Discriminant can be used to determine the number and type of roots of a quadratic equation.

Discriminant Value of Discriminant Type and Number of Roots b 2 – 4 ac > 0; b 2 – 4 ac is a perfect square. 2 real, rational roots (can be written as an integer or fraction) b 2 – 4 ac > 0; 2 real, irrational roots b 2 – 4 ac is not a perfect (cannot be written as square. an integer) b 2 – 4 ac = 0 b 2 – 4 ac < 0 (negative) 1 real, rational root (at the vertex) NO REAL ROOTS (2 complex roots) Example of Graph of Related Function

Describe Roots Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. • A. The discriminant is 0, so there is one rational root. • B. The discriminant is negative, so there are NO REAL ROOTS.

Describe Roots • Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. • C. The discriminant is 44, which is not a perfect square. Therefore there are two irrational roots. D. The discriminant is 289, which is a perfect square. Therefore, there are two rational roots.

MORE EXAMPLES

Example 1 Use the quadratic formula to solve the equation : x 2 + 5 x + 6= 0 Solution: x 2 + 5 x + 6= 0 a=1 b=5 c=6 x = - 2 or x = - 3 These are the roots of the equation.

Example 2 Use the quadratic formula to solve the equation : 8 x 2 + 2 x - 3= 0 Solution: 8 x 2 + 2 x - 3= 0 a = 8 b = 2 c = -3 x = ½ or x = - ¾ These are the roots of the equation.

Example 3 Use the quadratic formula to solve the equation : 8 x 2 - 22 x + 15= 0 Solution: 8 x 2 - 22 x + 15= 0 a = 8 b = -22 c = 15 x = 3/2 or x = 5/4 These are the roots of the equation.

Example 4 Use the quadratic formula to solve for x 2 x 2 +3 x - 7= 0 Solution: 2 x 2 + 3 x – 7 = 0 a=2 b=3 c=-7 These are the roots of the equation.