Guided Learning Extension: Factor over the Integers, Real and Complex number sets
We learned about factoring as a method for solving quadratic equations. In this extension, we're going to see how we can solve in order to factor.
Example 1: Factor over the Integers: \(40x^2-114x-135\).
\(\begin{align}&x=\frac{684\pm\sqrt{684^2-4\left(240\right)\left(-810\right)}}{2\left(240\right)}\ & \ \ \ &\text{1) Substitute a, b, and c values into the quadratic formula.}\\
&x=\frac{684\pm\sqrt{1245456}}{480}\ & \ \ \ &\text{2) Simplify.}\\
&x=\frac{684+1116}{480};\ \ x=\frac{684-1116}{480}\ & \ \ \ &\text{3) Write two equations.}\\
&x=\frac{15}{4};\ \ x=-\frac{9}{10}\ & \ \ \ &\text{4) Solve for x.}\\
&40(x-\frac{15}{4})(x+\frac{9}{10})\ & \ \ \ &\text{5) Write the factors with the solutions you found.}\\
&\ &\ &\text{Note that you need to include the a-value of the original equation.}\\
&4(x-\frac{15}{4}){\cdot}10(x+\frac{9}{10})\ & \ \ \ &\text{6) To get integers, you want to distribute the a-value so that there are no fractions.}\\
&\ &\ &\text{Here, multiplying the first quantity by 4 and the second by 10 eliminates the fractions.}\\
&(4x-15)(10x+9)\ & \ \ \ &\text{7) Simplify.}\end{align}\)
Example 2: Factor over \(\mathbb{R}\): \(9x^2+12x-1\).
\(\begin{align}&x=\frac{-12\pm\sqrt{12^2-4(9)(-1)}}{2\left(9\right)}\ & \ \ \ &\text{1) Substitute a, b, and c values into the quadratic formula.}\\
&x=\frac{-12\pm\sqrt{180}}{18}\ & \ \ \ &\text{2) Simplify.}\\
&x=\frac{-12+6\sqrt{5}}{18};\ \ x= \frac{-12-6\sqrt{5}}{18}\ & \ \ \ &\text{3) Write two equations.}\\
&x=\frac{-2+\sqrt{5}}{3};\ \ x= \frac{-2-\sqrt{5}}{3}\ & \ \ \ &\text{4) Solve for x.}\\
&9\left(x-\frac{-2+\sqrt{5}}{3}\right)\left(x- \frac{-2-\sqrt{5}}{3}\right)\ & \ \ \ &\text{5) Write the factors with the solutions you found.}\\
&\ &\ &\text{Note that you need to include the a-value of the original equation.}\\
&9\left(x+\frac{2-\sqrt{5}}{3}\right)\left(x+\frac{2+\sqrt{5}}{3}\right)\ & \ \ \ &\text{6) Simplify the quantities.}\\
&3\left(x+\frac{2-\sqrt{5}}{3}\right){\cdot}3\left(x+\frac{2+\sqrt{5}}{3}\right)\ & \ \ \ &\text{7) Distribute the a-value if necessary.}\\
&\left(3x+2-\sqrt{5}\right)\cdot\left(3x+2+\sqrt{5}\right)\ & \ \ \ &\text{8) Simplify.}\end{align}\)
Example 3: Factor over \(\mathbb{C}\): \(x^2+11\).
\(\begin{align}&x^2+11=0\ & \ \ \ &\text{1) Set up an equation to solve for x.}\\
&x^2=-11\ & \ \ \ &\text{2) Subtract.}\\
&x=\pm\sqrt{11}i\ & \ \ \ &\text{3) Evaluate the square root.}\\
&(x+\sqrt{11}i)(x-\sqrt{11}i)\ & \ \ \ &\text{4) Write factors with your solutions. Check to see if there is an a-value other than 1.}\end{align}\)
Guided Learning
We learned about factoring as a method for solving quadratic equations. In this extension, we're going to see how we can solve in order to factor.
Example 1: Factor over the Integers: \(40x^2-114x-135\).
\(\begin{align}&x=\frac{684\pm\sqrt{684^2-4\left(240\right)\left(-810\right)}}{2\left(240\right)}\ & \ \ \ &\text{1) Substitute a, b, and c values into the quadratic formula.}\\
&x=\frac{684\pm\sqrt{1245456}}{480}\ & \ \ \ &\text{2) Simplify.}\\
&x=\frac{684+1116}{480};\ \ x=\frac{684-1116}{480}\ & \ \ \ &\text{3) Write two equations.}\\
&x=\frac{15}{4};\ \ x=-\frac{9}{10}\ & \ \ \ &\text{4) Solve for x.}\\
&40(x-\frac{15}{4})(x+\frac{9}{10})\ & \ \ \ &\text{5) Write the factors with the solutions you found.}\\
&\ &\ &\text{Note that you need to include the a-value of the original equation.}\\
&4(x-\frac{15}{4}){\cdot}10(x+\frac{9}{10})\ & \ \ \ &\text{6) To get integers, you want to distribute the a-value so that there are no fractions.}\\
&\ &\ &\text{Here, multiplying the first quantity by 4 and the second by 10 eliminates the fractions.}\\
&(4x-15)(10x+9)\ & \ \ \ &\text{7) Simplify.}\end{align}\)
Example 2: Factor over \(\mathbb{R}\): \(9x^2+12x-1\).
\(\begin{align}&x=\frac{-12\pm\sqrt{12^2-4(9)(-1)}}{2\left(9\right)}\ & \ \ \ &\text{1) Substitute a, b, and c values into the quadratic formula.}\\
&x=\frac{-12\pm\sqrt{180}}{18}\ & \ \ \ &\text{2) Simplify.}\\
&x=\frac{-12+6\sqrt{5}}{18};\ \ x= \frac{-12-6\sqrt{5}}{18}\ & \ \ \ &\text{3) Write two equations.}\\
&x=\frac{-2+\sqrt{5}}{3};\ \ x= \frac{-2-\sqrt{5}}{3}\ & \ \ \ &\text{4) Solve for x.}\\
&9\left(x-\frac{-2+\sqrt{5}}{3}\right)\left(x- \frac{-2-\sqrt{5}}{3}\right)\ & \ \ \ &\text{5) Write the factors with the solutions you found.}\\
&\ &\ &\text{Note that you need to include the a-value of the original equation.}\\
&9\left(x+\frac{2-\sqrt{5}}{3}\right)\left(x+\frac{2+\sqrt{5}}{3}\right)\ & \ \ \ &\text{6) Simplify the quantities.}\\
&3\left(x+\frac{2-\sqrt{5}}{3}\right){\cdot}3\left(x+\frac{2+\sqrt{5}}{3}\right)\ & \ \ \ &\text{7) Distribute the a-value if necessary.}\\
&\left(3x+2-\sqrt{5}\right)\cdot\left(3x+2+\sqrt{5}\right)\ & \ \ \ &\text{8) Simplify.}\end{align}\)
Example 3: Factor over \(\mathbb{C}\): \(x^2+11\).
\(\begin{align}&x^2+11=0\ & \ \ \ &\text{1) Set up an equation to solve for x.}\\
&x^2=-11\ & \ \ \ &\text{2) Subtract.}\\
&x=\pm\sqrt{11}i\ & \ \ \ &\text{3) Evaluate the square root.}\\
&(x+\sqrt{11}i)(x-\sqrt{11}i)\ & \ \ \ &\text{4) Write factors with your solutions. Check to see if there is an a-value other than 1.}\end{align}\)
Guided Learning