Chapter 1.8, Problem 30E

Solution

Solve the equation $9n^2-42n-162=21n$ by quadratic formula and then by factoring to check solution.

The solution for the quadratic equation is $n=\left\{-2,9\right\}$ .

Given:

The quadratic equation is $9n^2-42n-162=21n$ .

Concept used:

Any quadratic equation of the form $a{x}^2+bx+c=0\text{ where }a\ne 0$ has the roots $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{......}\left(1\right)$

Calculation:

The quadratic equation $9n^2-42n-162=21n$ is re-written as:

  $\begin{array}{c}9n^2-42n-162=21n\mathrm{......}\left(\text{Given}\right)\\ 9n^2-63n-162=0\\ n^2-7n-18=0\mathrm{......}\left(2\right)\end{array}$

The solution for the quadratic equation $\left(2\right)$ can be evaluated with the help of formula $\left(1\right)$ . Therefore,

  $\begin{array}{c}n=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{......}\left(\text{From }\left(1\right)\right)\\ =\frac{-\left(-7\right)\pm \sqrt{{\left(-7\right)}^2-4\left(1\right)\left(-18\right)}}{2\left(1\right)}\mathrm{......}\left(\because n^2-7n-18=0\right)\\ =\frac{7\pm \sqrt{121}}{2}\\ =\frac{7+11}{2},\frac{7-11}{2}\\ =\left\{9,-2\right\}\end{array}$

Hence the solution for the quadratic equation is $n=\left\{9,-2\right\}$ .

Now, factoring the equation $\left(2\right)$ gives,

  $\begin{array}{c}{n}^2-7n-18=0\mathrm{......}\left(\text{From }\left(2\right)\right)\\ n^2-9n+2n-18=0\\ n\left(n-9\right)+2\left(n-9\right)=0\\ \left(n+2\right)\left(n-9\right)=0\\ n+2=0\text{ or }n-9=0\\ n=-2\text{ or }n=9\\ n=\left\{-2,9\right\}\end{array}$

From both the techniques the solutions are same for of the quadratic equation. Hence the required solution is $n=\left\{-2,9\right\}$ of quadratic equation.