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**Solving Quadratic Equations**

…by Factoring

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**Solving Quadratic Equations by Factoring**

Get ZERO on one side by itself. Factor. Consider Common Factors FIRST! Set each factor = 0. Solve each part.

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**Solve the second equation.**

Solve by factoring. Original equation Add 4x to each side. Factor the binomial. Zero Product Property or Solve the second equation. Answer: The solution set is {0, –4}. Example 3-1a

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**Check Substitute 0 and –4 in for x in the original equation.**

Example 3-1a

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**Subtract 5x and 2 from each side.**

Solve by factoring. Original equation Subtract 5x and 2 from each side. Factor the trinomial. Zero Product Property or Solve each equation. Answer: The solution set is Check each solution. Example 3-1a

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**Solve each equation by factoring. a.**

Answer: {0, 3} Answer: Example 3-1b

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**Answer: The solution set is {3}.**

Solve by factoring. Original equation Add 9 to each side. Factor. Zero Product Property or Solve each equation. Answer: The solution set is {3}. Example 3-2a

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Check The graph of the related function, intersects the x-axis only once. Since the zero of the function is 3, the solution of the related equation is 3. Example 3-2a

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Solve by factoring. Answer: {–5} Example 3-2b

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**Multiple-Choice Test Item What is the positive solution of the equation ?**

B 5 C 6 D 7 Read the Test Item You are asked to find the positive solution of the given quadratic equation. This implies that the equation also has a solution that is not positive. Since a quadratic equation can either have one, two, or no solutions, we should expect this equation to have two solutions. Example 3-3a

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**Solve the Test Item Original equation Factor. Divide each side by 2.**

Zero Product Property Solve each equation. Both solutions, –3 and 7, are listed among the answer choices. However, the question asks for the positive solution, 7. Answer: D Example 3-3a

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**Multiple-Choice Test Item What is the positive solution of the equation ?**

B –5 C 2 D 6 Answer: C Example 3-3b

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**Write a quadratic equation with and 6 as its **

roots. Write the equation in the form where a, b, and c are integers. Write the pattern. Replace p with and q with 6. Simplify. Example 3-4a

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**Multiply each side by 3 so that b is an integer.**

Use FOIL. Multiply each side by 3 so that b is an integer. Answer: A quadratic equation with roots and 6 and integral coefficients is You can check this result by graphing the related function. Example 3-4a

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**Write a quadratic equation with and 5 as its **

roots. Write the equation in the form where a, b, and c are integers. Answer: Example 3-4b

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EXAMPLE 1 Solve a quadratic equation having two solutions Solve x 2 – 2x = 3 by graphing. STEP 1 Write the equation in standard form. Write original equation.

EXAMPLE 1 Solve a quadratic equation having two solutions Solve x 2 – 2x = 3 by graphing. STEP 1 Write the equation in standard form. Write original equation.

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