Question

Factories the given equation : $7{p^2} - 5p = 2$

Answer 1

Hint: In the given question we will use factorization method to factories this. 1st we will make the right hand side 0 by subtracting 2 from both sides then the equation will become $7{p^2} - 5p - 2 = 0$. For factorization, split 5 in two parts such as the product of 2 parts will be equal to product 7 and -2. It will become $7{p^2} - 7p + 2p - 2 = 0$ after taking common and equating with 0 we will get the value of p.

Complete step-by-step answer:
$7{p^2} - 5p = 2$
Subtracting 2 from both side we get
$\Rightarrow$ $7{p^2} - 5p - 2 = 0$
Split 5 in two part such as the product of 2 parts will be equal to product 7 and -2
$\Rightarrow$ $7{p^2} - 7p + 2p - 2 = 0$
Taking common 7p and 2 we get
$ \Rightarrow 7p\left( {p - 1} \right) + 2\left( {p - 1} \right) = 0$
Taking $\left( {p - 1} \right)$ we get
\[ \Rightarrow \left( {p - 1} \right)\left( {7p + 2} \right) = 0\]
If \[\left( {p - 1} \right) = 0\] then $p = 1$
And \[\left( {7p + 2} \right) = 0\]then $p = \dfrac{{ - 2}}{7}$
Therefore $p = 1,\dfrac{{ - 2}}{7}$.

Note: Alternative method of finding root
$7{p^2} - 5p - 2 = 0$ where $a = 7,b = - 5,c = - 2$
Roots of quadratic equation $ = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
$ \Rightarrow \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - \left( { - 5} \right) \pm \sqrt {{{\left( { - 5} \right)}^2} - 4 \times 7 \times \left( { - 2} \right)} }}{{2\left( { - 2} \right)}}$
$ \Rightarrow \dfrac{{5 \pm \sqrt {25 + 56} }}{{ - 4}}$
$ \Rightarrow \dfrac{{5 \pm 9}}{{ - 4}}$
$\therefore p = 1,\dfrac{{ - 2}}{7}$