Question

How do you factor $2{{m}^{2}}-11m+15$?

Answer 1

Hint: For this problem they have asked to calculate the factors of the given equation. We can observe that the given equation is a quadratic equation which is in terms of $m$. We will calculate the roots of the given equation by using the quadratic formula which is $m=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. We will compare the given equation with the standard form of quadratic equation to get the values of $a$, $b$, $c$. After getting these values, we will use the quadratic formula and calculate the roots of the equation. After getting the roots of the given equation as $\alpha $, $\beta $, we can write the factors of the given equation as $m-\alpha $, $m-\beta $.

Complete step by step solution:
Given that, $2{{m}^{2}}-11m+15$.
Comparing the given equation with the standard form of the equation which is $a{{m}^{2}}+bm+c$, then we will get
$a=2$, $b=-11$, $c=15$.
Now the roots of the given equation from the quadratic formula are
$\Rightarrow m=\dfrac{-\left( -11 \right)\pm \sqrt{{{\left( -11 \right)}^{2}}-4\left( 2 \right)\left( 15 \right)}}{2\left( 2 \right)}$
When we multiply a negative sign with the negative sign, then we will get a positive sign as a result. Applying this rule in the above equation, then we will get
$\begin{align}
  & \Rightarrow m=\dfrac{11\pm \sqrt{121-120}}{4} \\
 & \Rightarrow m=\dfrac{11\pm \sqrt{1}}{4} \\
\end{align}$
In the above equation we have the value $\sqrt{1}$. We know that the value of $\sqrt{1}=1$. Substituting this value in the above equation, then we will get
$\Rightarrow m=\dfrac{11\pm 1}{4}$
Simplifying the above equation, then we will have
$\begin{align}
  & \Rightarrow m=\dfrac{11+1}{4}\text{ or }\dfrac{11-1}{4} \\
 & \Rightarrow m=\dfrac{12}{4}\text{ or }\dfrac{10}{4} \\
 & \Rightarrow m=3\text{ or }\dfrac{5}{2} \\
\end{align}$
We have the roots of the given equation as $3$, $\dfrac{5}{2}$.
Hence the factors for the given equation $2{{m}^{2}}-11m+15$ are $m-3$, $m-\dfrac{5}{2}$.

Note: We can also follow another method to find the factors of the given equation, which is a part of the factorization method. In this method we will split the middle term which is a coefficient of $m$ as a sum of two variables such that the product of that variables should be equal to product of coefficient of ${{m}^{2}}$ and $c$. After that we will take the appropriate terms as common from the equation to get the factors.