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Section 8.4 Quadratic Formula

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**Objectives Solving Quadratic Equations The Discriminant**

Quadratic Equations Having Complex Solutions

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QUADRATIC FORMULA The solutions to ax2 + bx + c = 0 with a ≠ 0 are given by

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Example Solve the equation 4x2 + 3x – 8 = 0. Support your results graphically. Solution Symbolic Solution Let a = 4, b = 3 and c = − 8. or or

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Example (cont) 4x2 + 3x – 8 = 0 Graphical Solution

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Example Solve the equation 3x2 − 6x + 3 = 0. Support your result graphically. Solution Let a = 3, b = −6 and c = 3.

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Example Solve the equation 2x2 + 4x + 5 = 0. Support your result graphically. Solution Let a = 2, b = 4 and c = 5. There are no real solutions for this equation because is not a real number.

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**THE DISCRIMINANT AND QUADRATIC**

EQUATIONS To determine the number of solutions to the quadratic equation ax2 + bx + c = 0, evaluate the discriminant b2 – 4ac. 1. If b2 – 4ac > 0, there are two real solutions. 2. If b2 – 4ac = 0, there is one real solution. 3. If b2 – 4ac < 0, there are no real solutions; there are two complex solutions.

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Example Use the discriminant to determine the number of solutions to −2x2 + 5x = 3. Then solve the equation using the quadratic formula. Solution −2x2 + 5x − 3 = 0 Let a = −2, b = 5 and c = −3. b2 – 4ac = (5)2 – 4(−2)(−3) = 1 or Thus, there are two solutions.

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THE EQUATION x2 + k = 0 If k > 0, the solution to x2 + k = 0 are given by

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Example Solve x = 0. Solution The solutions are

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Example Solve 3x2 – 7x + 5 = 0. Write your answer in standard form: a + bi. Solution Let a = 3, b = −7 and c = 5. and

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Example Solve Write your answer in standard form: a + bi. Solution Begin by adding 2x to each side of the equation and then multiply by 5 to clear fractions. Let a = −2, b = 10 and c = −15.

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Example (cont) Let a = −2, b = 10 and c = −15.

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Example Solve by completing the square. Solution After applying the distributive property, the equation becomes Since b = −4 ,add to each side of the equation. The solutions are 2 + i and 2 − i.

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The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.

The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.

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