Solve Quadratic Equations by Factoring
In general, solving a quadratic equation by factoring will be the most efficient if the equation is factorable. This is a method you are already familiar with as you have used it to graph quadratic functions in intercept form. Let’s look at two functions below and relate it to solving.
In general, solving a quadratic equation by factoring will be the most efficient if the equation is factorable. This is a method you are already familiar with as you have used it to graph quadratic functions in intercept form. Let’s look at two functions below and relate it to solving.
\(y=x^2-2x-15\) and \(y=\left(x+3\right)\left(x-5\right)\) are equivalent functions with \(x\)-intercepts at \(\left(-3,0\right)\) and \(\left(5,0\right)\). \(x=-3\) and \(x=5\) are values that correspond with \(y=0\).
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In the same way, \(y=-2x^2+6x\) and \(y=-2x\left(x-3\right)\) are equivalent functions with \(x\)-intercepts at \(\left(0,0\right)\) and \(\left(3,0\right)\). \(x=0\) and \(x=3\) are values that correspond with \(y=0\).
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If an equation is set equal to \(0\) and the expression is factorable, then we can solve it by setting each of the factors equal to \(0.\) To review factoring, go to Target B: Factor Quadratic Expressions Completely.
Steps to Solving Quadratic Equations by Factoring:
- If necessary, write an equivalent equation that is set equal to \(0\).
- Factor the quadratic expression.
- Set each factor equal to \(0\) and solve. You can check solutions by graphing or substituting in to original equation.
Note: If the expression is not factorable, then you can use one of the other three methods included in this target.
Solve the following equations.
Example 1: Solve \(3x^2-7=4x\).
\(\begin{align}&3x^2-4x-7=0\ & \ \ \ & \text{1) Set equation equal to 0.}\\&\left(3x-7\right)\left(x+1\right)=0\ & \ \ \ & \text{2) Factor.}\\&3x-7=0\ \text{and}\ x+1=0\ & \ \ \ & \text{3) Set each factor equal to 0.}\\&x=\frac{7}{3},-1\ & \ \ \ & \text{4) Solve.}\end{align}\)
Example 2: Solve \(20x=6x^2\).
\(\begin{align}&6x^2-20x=0\ & \ \ \ & \text{1) set equation equal to 0.}\\&2x(3x-10)=0\ & \ \ \ & \text{2) Factor.}\\&2x=0\ \text{and}\ \ 3x-10=0\ & \ \ \ & \text{3) Set each factor equal to 0.}\\&x=0,\frac{10}{3}\ & \ \ \ & \text{4) Solve.}\end{align}\)
Example 3: Solve \(-5x^2+3x=-2\).
Quick Check
Solve the following equations:
1) \(x^2-11x+24=0\)
2) \(9x^2-25=0\)
3) \(2x^2-7=13x\)
Quick Check Solutions
Solve the following equations:
1) \(x^2-11x+24=0\)
2) \(9x^2-25=0\)
3) \(2x^2-7=13x\)
Quick Check Solutions