Question

How do you factorize $2{{p}^{2}}+2p-4$?

Answer 1

Hint: Now to factorize the expression we will first find the roots of the expression. Now to find the roots we will first divide the equation by 2 to make the coefficient of ${{p}^{2}}$ as 1 . Now we will add and subtract the term \[{{\left( \dfrac{b}{2a} \right)}^{2}}\] in the equation and then simplify the equation with the help of formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ . Now we will simplify the equation and find the roots of the equation. Now we know that if $\alpha $ and $\beta $ are the roots of a quadratic equation then $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ are the factors of the expression.

Complete step by step solution:
Now to factorize the given expression let us consider the equation $2{{p}^{2}}+2p-4=0$ .
First we will find the roots of the equation with the help of completing the square method.
Now let us compare the expression $2{{p}^{2}}+2p-4$ with $a{{x}^{2}}+bx+c=0$.
Hence we get, a = 2, b = 2, c = - 4.
Now to solve the expression by complete square method we need a = 1.
Hence we will divide the whole equation by 2. Hence we get,
${{p}^{2}}+p-2=0$
Now we want to make a perfect square in the equation. Hence we will add and subtract the equation with ${{\left( \dfrac{b}{2a} \right)}^{2}}$ .
Hence we get, ${{p}^{2}}+p+\dfrac{1}{4}-\dfrac{1}{4}-2=0$
Now we know that ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$
Hence we can write the equation as,
$\begin{align}
  & \Rightarrow {{\left( p+\dfrac{1}{2} \right)}^{2}}-\dfrac{1}{4}-2=0 \\
 & \Rightarrow {{\left( p+\dfrac{1}{2} \right)}^{2}}=2+\dfrac{1}{4} \\
 & \Rightarrow {{\left( p+\dfrac{1}{2} \right)}^{2}}=\dfrac{9}{4} \\
\end{align}$
Now taking square root on both sides we get,
$\Rightarrow p+\dfrac{1}{2}=\pm \dfrac{3}{2}$
Now taking $\dfrac{1}{2}$ on RHS we get,
$\Rightarrow p=\dfrac{3}{2}-\dfrac{1}{2}$ or $p=-\dfrac{3}{2}-\dfrac{1}{2}$
Hence we get $p=\dfrac{3-1}{2}$ or $\dfrac{-3-1}{2}$
Hence we have p = 1 or p = - 2.
Now we know that if $\alpha $ and $\beta $ are the roots of a quadratic equation then $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ are the factors of the expression.
Hence we have $\left( x-1 \right)$ and $\left( x+2 \right)$ as factors of the expression $2{{p}^{2}}+2p-4$

Note: Now to find roots we can also use the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ which gives roots to the quadratic equation. We can also directly factorize the equation. To do so we must first split the middle terms in such a way that the product of the terms is equal to multiplication of first term × last terms.