Question

How do you factor the expression \[133+208y+64{{y}^{2}}\]?

Answer 1

Hint: Now we are given with the quadratic expression in y. We know that the roots of the quadratic expression are given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence using this formula we will first find the roots of the quadratic expression. Now we know that if $\alpha $ is the root of the quadratic expression then $\left( x-\alpha \right)$ is the factor of the expression. Hence using this we will factor the given expression.

Complete step by step solution:
Now consider the given expression \[133+208y+64{{y}^{2}}\]
We know that the given expression is a quadratic expression in y of the form $a{{y}^{2}}+by+c$ where a = 64, b = 208 and c = 133.
Now we know that the roots of any quadratic expression of the form $a{{x}^{2}}+bx+c$ is given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Hence first let us find the roots of the given expression by substituting the values of a, b and c.
Hence the roots of the expression are given by,
\[\begin{align}
  & \Rightarrow y=\dfrac{-208\pm \sqrt{{{208}^{2}}-4\left( 133 \right)\left( 64 \right)}}{2\left( 133 \right)} \\
 & \Rightarrow y=\dfrac{-208\pm \sqrt{9216}}{266} \\
 & \Rightarrow y=\dfrac{-208\pm 96}{266} \\
\end{align}\]
Hence the value of y is \[y=\dfrac{-208+96}{266}\] or $y=\dfrac{-208-96}{266}$
Hence we get the roots of the given expression are $\dfrac{-112}{266}$ and $\dfrac{-304}{266}$ .
Now we know that if $\alpha $ and $\beta $ are the roots of the expression then $x-\alpha $ and $x-\beta $ are the factors of the given expression.
Hence we get the factors of the expression given are $\left( x+\dfrac{112}{266} \right)$ and $\left( x+\dfrac{304}{266} \right)$

Hence the factorization of the expression given will be \[\left( x+\dfrac{112}{266} \right)\left( x+\dfrac{304}{266} \right)\] .

Note: Now we can find the roots by using completing square method also. In this method we first make the coefficient of ${{x}^{2}}$ as 1. Then we will add and subtract the term ${{\left( \dfrac{b}{2a} \right)}^{2}}$ and use the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ . Hence using this we will simplify the expression and solve to find the roots of the expression.