Chapter 1.8, Problem 27E

Solution

Solve the equation $3x^2+7x-24=13x$ by quadratic formula and then by factoring to check solution.

The solution for the quadratic equation is $x=\left\{-2,4\right\}$ .

Given:

The quadratic equation is $3x^2+7x-24=13x$ .

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 $3x^2+7x-24=13x$ is re-written as:

  $\begin{array}{c}3x^2+7x-24=13x\mathrm{......}\left(\text{Given}\right)\\ 3x^2-6x-24=0\mathrm{......}\left(2\right)\end{array}$

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

  $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{......}\left(\text{From }\left(1\right)\right)\\ =\frac{-\left(-6\right)\pm \sqrt{{\left(-6\right)}^2-4\left(3\right)\left(-24\right)}}{2\left(3\right)}\mathrm{......}\left(\because 3x^2-6x-24=0\right)\\ =\frac{6\pm \sqrt{324}}{6}\\ =\frac{6+18}{6},\frac{6-18}{6}\\ =\left\{4,-2\right\}\end{array}$

Hence the solution for the quadratic equation is $s=\left\{3,-1\right\}$ .

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

  $\begin{array}{c}3x^2-6x-24=0\mathrm{......}\left(\text{From }\left(2\right)\right)\\ 3x^2-12x+6x-24=0\\ 3x\left(x-4\right)+6\left(x-4\right)=0\\ \left(3x+6\right)\left(x-4\right)=0\\ 3x+6=0\text{ or }x-4=0\\ x=-2\text{ or }x=4\\ x=\left\{-2,4\right\}\end{array}$

From both the techniques the solutions are same for of the quadratic equation. Hence the required solution is $x=\left\{-2,4\right\}$ of quadratic equation.