solving quadratic equations pulling it all together
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Solving Quadratic EquationsPulling It All Together
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Five ways to solve…
• Factoring
• Square Root Principle
• Completing the Square
• Quadratic Formula (not this test, test 3)
• Graphing
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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 = 0x = -2 x = -3
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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
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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.
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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.
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Solve by Graphing: x2 + 5x + 6 = 0
x = -3 x = -2
1. Enter the polynomial into the “y=“ function of the calculator.
2. Modify the window as needed to accommodate the graph.
3. Locate the x-intercepts of the graph. These are the solutions to the equation.
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Graph these Quadratics X2 - 4 = 0 X2 - 4x + 4 = 0 X2 + 4x - 4 = 0
Based on the graphs for the equations above, what are thepossibilities for solutions to a quadratic equation?