Question

Factorize the following using appropriate identities.
$3{p^2} - 24p + 36$

Answer 1

Hint: In this question, we are given an equation and we have been asked to factorize the equation using appropriate identities. Split the equation in such a way that it forms the identity of $\left( {x - a} \right)\left( {x - b} \right)$. Then, identify the $'a'$ and $'b'$, and form the formula. Make sure that the term ${p^2}$ has no coefficient. So, take the coefficient of ${p^2}$ common and then factorize the equation.

Complete step-by-step solution:
We are given an equation, which we have been asked to factorize.
But first, we need to make sure that the equation is in the form of ${x^2} - \left( {a + b} \right)x + ab$. In the equation that has been given to us,${p^2}$ has a coefficient with which is not required. So, we will take the common coefficient of ${p^2}$ from the entire equation.
$ \Rightarrow 3{p^2} - 24p + 36$
Now, we will take the common coefficient of ${p^2}$.
$ \Rightarrow 3\left( {{p^2} - 8p + 12} \right)$
Now, we have to find 2 such numbers which will give us 8 on being added and 12 on being multiplied. Two such numbers are $6$ and ${\text{2}}$. On simplifying we will get,
$ \Rightarrow 3\left[ {{p^2} - \left( {6 + 2} \right)p + \left( {6 \times 2} \right)} \right]$
We know the identity: ${x^2} - \left( {a + b} \right)x + ab$$ = \left( {x - a} \right)\left( {x - b} \right)$
As we can see, $a = 6,b = 2$
Putting in the identity, we get –
$ \Rightarrow 3\left( {p - 6} \right)\left( {p - 2} \right)$

Hence, the factors of $3{p^2} - 24p + 36$ are $3$, $(p-6)$ and $(p-2)$.

Note: We already known that, the sum of the roots of a quadratic equation is equal to the negation of coefficient of the second term, divided by the leading coefficient $\left( {{r_1} + {r_2}} \right) = - \dfrac{b}{a}$. The product of the roots of a quadratic equation is equal to the constant term (the third term), divided by the leading coefficient ${r_1} \cdot {r_2} = \dfrac{c}{a}$. Hence, the general quadratic equation of the form will be ${x^2} - \left( {a + b} \right)x + ab$.