Solving Quadratic Equations
Pulling It All Together
Five ways to solve…
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•
•
•
•
Factoring
Square Root Principle
Completing the Square
Quadratic Formula (not this test, test 3)
Graphing
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
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
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.
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.
Solve by Graphing:
x2 + 5x + 6 = 0
1. Enter the polynomial into
the “y=“ function of the
calculator.
2. Modify the window as
needed to accommodate
the graph.
x = -3
x = -2
3. Locate the x-intercepts of
the graph. These are the
solutions to the equation.
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?