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Solving Quadratic Equations Section 1.3. What is a Quadratic Equation? . A quadratic equation in x is an equation that can be written in the standard form: ax ² + bx + c = 0 Where a,b,and c are real numbers and a ≠ 0. Solving a Quadratic Equation by Factoring.
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What is a Quadratic Equation? A quadratic equationin 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.
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.
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)
Factor: 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 } Example 1Solve x² - 12x + 35 = 0 by factoring.
Example 2Solve 3t² + 10t + 6 = -2 by factoring. • Check equation to make sure it is in standard form before solving. Is it? • It is not, soset equation equal to zero first: 3t² + 10t + 8 = 0 • Now factor and solve. (3t + 4)(t + 2) = 0 3t + 4 = 0 t +2 = 0 t = t = -2
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).
Example 1:Solve by the Square Root Method: 2x² - 32 = 0 2x² = 32 x² = 16 = x = ± 4
Example 2:Solve by the Square Root Method. 5x² + 10 = 0 5x² = -10 x² = -2 x = ± x = ±i
Example 3:Solve by the Square Root Method. (x – 3)² = 25 x – 3 = ± 5 x – 3 = 5 or x – 3 = -5 x = 8 x = -2
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 Solve by Completing the Square.
x² + 8x – 3 = 0 x² + 8x = 3 x² + 8x + (4)² = 3 + (4)² x² + 8x + 16 = 3 + 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. Example 1:Solve by Completing the Square.
2x² - 4x + 3 = 0 x² - 2x + = 0 x² - 2x + ___ = + ____ x² - 2x + 1 = + 1 (x – 1)² = x – 1 = ± x = 1 ± Divide by the leading coefficient. Continue to solve using the completing the square method. Simplify radical. Example 2:Solve by Completing the Square when the Leading Coefficient is not equal to 1.
Quadratic Formula If a quadratic can’t be factored, you must use the quadratic formula. If ax² + bx + c = 0, then the solution is:
a = 1 b = -4 c = -1 Solve
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.
