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Mastering Quadratic Equations: Five Efficient Solution Methods

Learn efficient ways to solve quadratic equations using factoring, square root principle, completing the square, quadratic formula, and graphing. Step-by-step examples provided.

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Mastering Quadratic Equations: Five Efficient Solution Methods

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  1. Solving Quadratic EquationsPulling It All Together

  2. Five ways to solve… • Factoring • Square Root Principle • Completing the Square • Quadratic Formula (not this test, test 3) • Graphing

  3. Solve by Factoring: x2 + 5x + 6 = 0 • Get all the terms of the polynomial in descending order on one side of the equation and 0 on the other side. x2 + 5x + 6 = 0 • Factor the polynomial. (x + 2) (x + 3) = 0 • Apply the zero product rule by setting each factor equal to zero. x + 2 = 0 or x + 3 = 0 • Solve each equation for x. x + 2 = 0 or x + 3 = 0 x = -2 x = -3

  4. Solve using the Square Root Principle • Must have “perfect square” variable expression on one side and constant on the other Examples: x2 = 16 (x – 4)2 = 9 (2x – 1)2 = 5

  5. Solve by Completing the Square: x2 + 5x + 6 = 0 • Gather the x-terms to one side of the equation and the constant terms to the other side and simplify if possible. x2 + 5x = -6 • Divide the coefficient of x by 2, square the result, and add this number to both sides of the equation. x2 + 5x = -6 • Factor the polynomial and simplify the constants.

  6. Once the “Square is complete,”Apply the Square Root Principle • Take the square root of both sides (be sure to include plus/minus in front of the constant term). • Simplify both sides. • Solve for x.

  7. Solve by Graphing: x2 + 5x + 6 = 0 • Enter the polynomial into the “y=“ function of the calculator. • Modify the window as needed to accommodate the graph. • Locate the x-intercepts of the graph. These are the solutions to the equation. x = -3 x = -2

  8. Graph these Quadratics X2 - 4 = 0 X2 - 4x + 4 = 0 X2 + 4x - 4 = 0 Based on the graphs for the equations above, what are the possibilities for solutions to a quadratic equation?

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