Question

Is $m = - 10$ the root of the equation $\dfrac{{3m - 7}}{{8m + 1}} = 4 - m$ ?

Answer 1

Hint: Let $a{x^2} + bx + c = 0$ be a quadratic equation. $p$ is declared as a root of the equation $a{x^2} + bx + c = 0$ if $a{p^2} + bp + c = 0$ . A quadratic equation has two roots which may be real or imaginary. They may be equal or unequal. So we have to put the value of $m$ in the equation and check if both sides give us the same number. If they are equal then $m$ is the root of the equation else not.

Complete step-by-step answer:
We are given $m = - 10$ . We have to check if $m$ is the root of the equation $\dfrac{{3m - 7}}{{8m + 1}} = 4 - m$ .
If $m$ is a root of the equation then it will satisfy both sides of the equation and both sides of the equation should be equal.
First, we look into the left-hand side of the equation and we substitute the value of $m = - 10$ .
So, we get:
$\dfrac{{3m - 7}}{{8m + 1}}$
$ = \dfrac{{3 \times \left( { - 10} \right) - 7}}{{8 \times \left( { - 10} \right) + 1}}$
Simplifying both the numerator and denominator we get:
$ = \dfrac{{ - 30 - 7}}{{ - 80 + 1}}$
$ = \dfrac{{ - 37}}{{ - 79}}$
Canceling the negative sign from both numerator and denominator we get:
$\dfrac{{37}}{{79}}$


Next, we look into the left-hand side of the equation and we substitute the value of $m = - 10$ .
$4 - m$
$ = 4 - \left( { - 10} \right)$
We know that the negative of a negative number gives a positive number.
$ = 4 + 10$
Simplifying we get:
$ = 14$
Since $\dfrac{{37}}{{79}} \ne 14$ so $\dfrac{{3m - 7}}{{8m + 1}} \ne 4 - m$.
$\therefore m$ is not defined as a root of the equation $\dfrac{{3m - 7}}{{8m + 1}} = 4 - m$ .

Note: Another approach would be simplifying the quadratic equation by cross multiplication and find the roots of the quadratic \[a{x^2} + bx + c = 0\] using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ . If the roots of the quadratic match with the given root then we conclude that $m = - 10$ is a root of the equation $\dfrac{{3m - 7}}{{8m + 1}} = 4 - m$ and if not then $m = - 10$ is not a root of the equation.