The Quadratic Formula..

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Presentation transcript:

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 : ax2 + bx + c = 0 The roots of the quadratic equation are given by :

The Quadratic Formula The solutions of a quadratic equation of the form ax2 + bx + c = 0, where a ≠ 0, are given by the following formula:

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

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

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

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

Roots and the Discriminant The Discriminant = b2 – 4ac The value of the Discriminant can be used to determine the number and type of roots of a quadratic equation.

Discriminant b2 – 4ac > 0; b2 – 4ac is a perfect square. Value of Discriminant Type and Number of Roots Example of Graph of Related Function b2 – 4ac > 0; b2 – 4ac is a perfect square. 2 real, rational roots (can be written as an integer or fraction) b2 – 4ac > 0; b2 – 4ac is not a perfect square. 2 real, irrational roots (cannot be written as an integer) b2 – 4ac = 0 1 real, rational root (at the vertex) b2 – 4ac < 0 (negative) NO REAL ROOTS (2 complex roots)

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 + 5x + 6= 0 Solution: x 2 + 5x + 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 : 8x 2 + 2x - 3= 0 Solution: 8x 2 + 2x - 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 : 8x 2 - 22x + 15= 0 Solution: 8x 2 - 22x + 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 2x 2 +3x - 7= 0 Solution: 2x 2 + 3x – 7 = 0 a = 2 b = 3 c = - 7 These are the roots of the equation.