Chapter 1.8, Problem 8E

Solution

Solve the quadratic equation $5p^2-10p+24=0$ for $\left(p\right)$ .

The solution for the quadratic equation is $p=\left\{\frac{5+i\sqrt{95}}{5},\frac{5-i\sqrt{95}}{5}\right\}$ .

Given:

The quadratic equation is $5p^2-10p+24=0$ .

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 value of $\left(p\right)$ for the quadratic equation $5p^2-10p+24=0$ can be evaluated with the help of formula $\left(1\right)$ . Therefore,

  $\begin{array}{c}p=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{......}\left(\text{From }\left(1\right)\right)\\ =\frac{-\left(-10\right)\pm \sqrt{{\left(-10\right)}^2-4\left(5\right)\left(24\right)}}{2\left(5\right)}\mathrm{......}\left(\because 5p^2-10p+24=0\right)\\ =\frac{10\pm i2\sqrt{95}}{10}\\ =\frac{5\pm i\sqrt{95}}{5}\\ =\left\{\frac{5+i\sqrt{95}}{5},\frac{5-i\sqrt{95}}{5}\right\}\end{array}$

Hence the solution for the quadratic equation is $p=\left\{\frac{5+i\sqrt{95}}{5},\frac{5-i\sqrt{95}}{5}\right\}$ .