# How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given x^2-16x+4=0?

Jan 30, 2018

$\mathrm{di} s c r i \min a n t = \sqrt{240}$

Equation has two real roots

Two roots are $\textcolor{red}{x = 23.746 , 8.254}$

#### Explanation:

Given quadratic equation is ${x}^{2} - 16 x + 4 = 0$

Formula to find the roots $x = \left(- b \pm \frac{\sqrt{{b}^{2} - \left(4 a c\right)}}{2 a}\right)$

Where a, b, c are the coefficients of ${x}^{2} , x , c o n s t .$ terms respectively.

$a = 1 , b = - 16 , c = 4$

Term $\sqrt{{b}^{2} - \left(4 a c\right)}$ is called the discriminant.

If the discriminant term is

a) Positive - both the roots are real

b) Zero - one real solution

c) Negative - two complex solutions

$\mathrm{di} s c r i \min a n t = \sqrt{{\left(- 16\right)}^{2} - \left(4 \cdot 1 \cdot 4\right)} = \sqrt{240}$

Hence the equation has two real roots.

$x = - \left(- 16\right) \pm \frac{\sqrt{{\left(- 16\right)}^{2} - \left(4 \cdot 1 \cdot 4\right)}}{2 \cdot 1}$

$x = \frac{16 \pm \sqrt{256 - 16}}{2} = 8 \pm \frac{\sqrt{240}}{2}$

$x = 16 \pm \left(\frac{\cancel{2} \sqrt{60}}{\cancel{2}}\right)$

$x = 16 \pm 7.746$

$\textcolor{red}{x = 23.746 , 8.254}$