Question

How do you factor ${{m}^{2}}+2m-24$?

Answer 1

Hint: Now to factor the given equation we will first find the roots of the equation. Now we know that the roots of the quadratic equation are given by formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we will get the roots of the equation. Now we know that if $\alpha $ and $\beta $ are the roots of the equation then $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ are the factors of the equation. Hence we get the factors of the equation.

Complete step by step solution:
Now to find the factors of the given equation we will first find the roots of the equation.
Now we know the given equation is a quadratic equation in m.
Now the general quadratic equation in x is $a{{x}^{2}}+bx+c$ Now comparing the given equation with the general equation we get a = 1, b = 2 and c = - 24.
Now we know that the roots of quadratic equation $a{{x}^{2}}+bx+c$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Hence substituting the values of a, b and c in the equation we get,
$\begin{align}
  & \Rightarrow \dfrac{-2\pm \sqrt{{{2}^{2}}-4\left( 1 \right)\left( -24 \right)}}{2\left( 1 \right)} \\
 & \Rightarrow \dfrac{-2\pm \sqrt{4+69}}{2\left( 1 \right)} \\
 & \Rightarrow \dfrac{-2\pm 10}{2} \\
\end{align}$
Hence we get the roots of the equation are $\dfrac{-2+10}{2}=4$ and $\dfrac{-2-10}{2}=-6$ .
Hence the roots of the equation are 4 and – 6.
Now we know that if $\alpha $ and $\beta $ are roots of the equation then $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ are the factors of the equation.
Using his we get the factors of given equation are $\left( x-4 \right)$ and $\left( x-\left( -6 \right) \right)$
Hence we get the factors of the given equation are $\left( x-4 \right)$ and $\left( x+6 \right)$ .
Note: Now note that we can also factorize the equation by splitting the middle term method. In this method we split the middle term such that the product of the terms is equal to multiplication of the first term in the equation with the last term in the equation. Now we simplify the equation by taking common terms together and hence factorize the equation.