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§ 3.6

Solving Quadratic Equations by Factoring. § 3.6. Zero Factor Theorem. Quadratic Equations Can be written in the form ax 2 + bx + c = 0. a , b and c are real numbers and a  0. This is referred to as standard form . Zero Factor Theorem

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§ 3.6

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  1. Solving Quadratic Equations by Factoring § 3.6

  2. Zero Factor Theorem Quadratic Equations • Can be written in the form ax2 + bx + c = 0. • a, b and c are real numbers and a 0. • This is referred to as standard form. Zero Factor Theorem • If m and n are real numbers and mn = 0, then m = 0 or n = 0. • This theorem is very useful in solving quadratic equations.

  3. Solving Quadratic Equations Steps for Solving a Quadratic Equation by Factoring • Write the equation in standard form. • Factor the quadratic completely. • Set each factor containing a variable equal to 0. • Solve the resulting equations. • Check each solution in the original equation.

  4. Solving Quadratic Equations In general: ax2 + bx + c = 0.

  5. Solving Quadratic Equations Example 1 Solve x2 – 5x = 24. • First write the quadratic equation in standard form. x2 – 5x – 24 = 0 • Now we factor the quadratic using techniques from the previous sections. x2 – 5x – 24 = (x – 8)(x + 3) = 0 • We set each factor equal to 0. x – 8 = 0 or x + 3 = 0, which will simplify to x = 8 or x = – 3 Continued.

  6. Solving Quadratic Equations Example Continued • Check both possible answers in the original equation. 82 – 5(8) = 64 – 40 = 24 true (–3)2 – 5(–3) = 9 – (–15) = 24 true • So our solutions for x are 8 or –3.

  7. Solving Quadratic Equations 8x = 1 or 4x = – 5, which simplifies to x = or Example 2 Solve 4x(8x + 9) = 5 • First write the quadratic equation in standard form. 32x2 + 36x = 5 32x2 + 36x – 5 = 0 • Now we factor the quadratic using techniques from the previous sections. 32x2 + 36x – 5 = (8x – 1)(4x + 5) = 0 • We set each factor equal to 0. 8x – 1 = 0 or 4x + 5 = 0 Continued.

  8. Solving Quadratic Equations Put it all together: 4x(8x + 9) = 5 32x2 + 36x = 5 32x2 + 36x – 5 = 0 (8x – 1)(4x + 5) = 0 8x – 1 = 0 or 4x + 5 = 0

  9. Solving Quadratic Equations true true • So our solutions for x are or . Example Continued • Check both possible answers in the original equation.

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