# How do you find the value of the discriminant and determine the nature of the roots 8x^2 – 2x – 18 = -15 ?

Sep 4, 2016

The discriminant of the second degree equation $a {x}^{2} + b x + c = 0$is:
${b}^{2} - 4 \cdot a \cdot c$

and the solutions depend on whether the two values $\pm \sqrt{{b}^{2} - 4 \cdot a \cdot c}$, are real or not

#### Explanation:

The given equation in the standard form $a {x}^{2} + b x + c = 0$ is $8 {x}^{2} - 2 x - 3 = 0$

Then, $\pm \sqrt{{b}^{2} - 4 \cdot a \cdot c}$ in our case is $\pm \sqrt{{\left(- 2\right)}^{2} - 4 \cdot \left(8\right) \cdot \left(- 3\right)} = \pm \sqrt{100} = \pm 10$.

the value of the discriminant is 100, so the two values of the square root are real, and there are two real solutions to the equation.

Sep 4, 2016

There are 2 real roots which are irrational and distinct (different)

#### Explanation:

Make the quadratic equation = 0.

$8 {x}^{2} - 2 x - 3 = 0 \leftarrow$ this is in the form $a {x}^{2} + b x + c = 0$

The discriminant is given by $\Delta = {b}^{2} - 4 a c$

$\Delta = {\left(- 3\right)}^{2} - 4 \left(8\right) \left(- 3\right) = 9 + 96 = 105$

What does this value of 105 tell us?

$\Delta > 0 \rightarrow$ there are 2 distinct (different) real roots.

$\Delta = 105$ which is not a perfect square
$\rightarrow$ the roots are irrational.

There are 2 real roots which are irrational and distinct (different)