Chapter 11.7, Problem 30E

To determine

To describe:

The nature of the roots of the given discriminant.

Answer to Problem 30E

This is the $2$ imaginary roots and $x=\frac{4\pm 2i\sqrt{11}}{5}$ .

Explanation of Solution

Given:

The discriminant

  $5x^2-8x+12=0$

Concept used:

The number is the square root of the quadratic formula.

  $b^2-4ac$

The discriminant formula

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

The quadratic equation

  $5x^2-8x+12=\mathrm{0.......................}\left(1\right)$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{..................}\left(2\right)$

Comparing the quadratic equation $a{x}^2+bx+c=0$

  $a=5,b=-8,c=12$

  $\begin{array}{l}=b^2-4ac\\ ={\left(-8\right)}^2-4\left(5\right)\left(12\right)\\ =-176<0\end{array}$

The discriminant is negative.

So, this is $2$ imaginary roots.

Putting the value of $a,b,c$ in equation (2)

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-\left(-8\right)\pm \sqrt{{\left(-8\right)}^2-4\left(5\right)\left(12\right)}}{2\left(5\right)}\\ x=\frac{8\pm \sqrt{-176}}{10}\\ x=\frac{8\pm i\sqrt{16\times 11}}{10}\\ x=\frac{8\pm 4i\sqrt{11}}{10}\\ x=\frac{4\pm 2i\sqrt{11}}{5}\end{array}$

Therefore,

The value of exact solution is $x=\frac{4\pm 2i\sqrt{11}}{5}$ .