Chapter 1.8, Problem 41E

Solution

Solve the quadratic equation $7x-3x^2=85+2x^2+2x$ for $\left(x\right)$ by any method.

The solution for the quadratic equation is $x=\left\{\frac{1+i\sqrt{67}}{2},\frac{1-i\sqrt{67}}{2}\right\}$ .

Given:

The quadratic equation is $7x-3x^2=85+2x^2+2x$ which can be written as $5x^2-5x+85=0$ .

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 solution $\left(x\right)$ for the quadratic equation $5x^2-5x+85=0$ can be evaluated with the help of formula $\left(1\right)$ . Therefore,

  $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{......}\left(\text{From }\left(1\right)\right)\\ =\frac{-\left(-5\right)\pm \sqrt{{\left(-5\right)}^2-4\left(5\right)\left(85\right)}}{2\left(5\right)}\mathrm{......}\left(\because 5x^2-5x+85=0\right)\\ =\frac{5\pm i\sqrt{1675}}{10}\\ =\frac{5\pm 5i\sqrt{67}}{10}\\ =\left\{\frac{1+i\sqrt{67}}{2},\frac{1-i\sqrt{67}}{2}\right\}\end{array}$

Hence the solution for the quadratic equation is $x=\left\{\frac{1+i\sqrt{67}}{2},\frac{1-i\sqrt{67}}{2}\right\}$ .