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**Solving Quadratic Equations Section 1.3**

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**What is a Quadratic Equation?**

A quadratic equation in x is an equation that can be written in the standard form: ax² + bx + c = 0 Where a,b,and c are real numbers and a ≠ 0.

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**Solving a Quadratic Equation by Factoring.**

The factoring method applies the zero product property: Words: If a product is zero, then at least one of its factors has to be zero. Math: If (B)(C)=0, then B=0 or C=0 or both.

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**Recap of steps for how to solve by Factoring**

Set equal to 0 Factor Set each factor equal to 0 (keep the squared term positive) Solve each equation (be careful when determining solutions, some may be imaginary numbers)

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**Example 1 Solve x² - 12x + 35 = 0 by factoring.**

Set each factor equal to zero by the zero product property. Solve each equation to find solutions. The solution set is: (x – 7)(x - 5) = 0 (x – 7)=0 (x – 5)=0 x = 7 or x = 5 { 5, 7 }

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**Example 2 Solve 3t² + 10t + 6 = -2 by factoring.**

Check equation to make sure it is in standard form before solving. Is it? It is not, so set equation equal to zero first: 3t² + 10t + 8 = 0 Now factor and solve. (3t + 4)(t + 2) = 0 3t + 4 = t +2 = 0 t = t = -2

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Solve by factoring.

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**Solve by the Square Root Method.**

If the quadratic has the form ax² + c = 0, where a ≠ 0, then we could use the square root method to solve. Words: If an expression squared is equal to a constant, then that expression is equal to the positive or negative square root of the constant. Math: If x² = c, then x = ±c. Note: The variable squared must be isolated first (coefficient equal to 1).

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**Example 1: Solve by the Square Root Method:**

= x = ± 4

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**Example 2: Solve by the Square Root Method.**

x = ±i

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**Example 3: Solve by the Square Root Method.**

x – 3 = 5 or x – 3 = -5 x = x = -2

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**Solve by the Square Root Method**

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**Solve by Completing the Square.**

Words Express the quadratic equation in the following form. Divide b by2 and square the result, then add the square to both sides. Write the left side of the equation as a perfect square. Solve by using the square root method. Math x² + bx = c x² + bx + ( )² = c + ( )² (x )² = c + ( )²

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**Example 1: Solve by Completing the Square.**

x² + 8x – 3 = 0 x² + 8x = 3 x² + 8x + (4)² = 3 + (4)² x² + 8x + 16 = (x + 4)² = 19 x + 4 = ± x = -4 ± Add three to both sides. Add ( )² which is (4)² to both sides. Write the left side as a perfect square and simplify the right side. Apply the square root method to solve. Subtract 4 from both sides to get your two solutions.

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**Example 2: Solve by Completing the Square when the Leading Coefficient is not equal to 1.**

2x² - 4x + 3 = 0 x² - 2x = 0 x² - 2x + ___ = ____ x² - 2x + 1 = (x – 1)² = x – 1 = ± x = 1 ± Divide by the leading coefficient. Continue to solve using the completing the square method. Simplify radical.

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Quadratic Formula If a quadratic can’t be factored, you must use the quadratic formula. If ax² + bx + c = 0, then the solution is:

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a = 1 b = -4 c = -1 Solve

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Solve

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Solve

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Discriminant The term inside the radical b² - 4ac is called the discriminant. The discriminant gives important information about the corresponding solutions or answers of ax² + bx + c = 0, where a,b, and c are real numbers. b² - 4ac Solutions b² - 4ac > 0 b² - 4ac = 0 b² - 4ac < 0

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**Tell what kind of solution to expect**

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