# What is the discriminant of 5x^2-8x-3=0 and what does that mean?

Jul 24, 2015

The discriminant of an equation tells the nature of the roots of a quadratic equation given that a,b and c are rational numbers.

$D = 124$

#### Explanation:

The discriminant of a quadratic equation $a {x}^{2} + b x + c = 0$ is given by the formula ${b}^{2} + 4 a c$ of the quadratic formula;

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The discriminant actually tells you the nature of the roots of a quadratic equation or in other words, the number of x-intercepts, associated with a quadratic equation.

Now we have an equation;

5x^2−8x−3=0

Now compare the above equation with quadratic equation $a {x}^{2} + b x + c = 0$, we get $a = 5 , b = - 8 \mathmr{and} c = - 3$.

Hence the discriminant (D) is given by;

$D = {b}^{2} - 4 a c$
$\implies D = {\left(- 8\right)}^{2} - 4 \cdot 5 \cdot \left(- 3\right)$
$\implies D = 64 - \left(- 60\right)$
$\implies D = 64 + 60 = 124$

Therefore the discriminant of a given equation is 124.

Here the discriminant is greater than 0 i.e. ${b}^{2} - 4 a c > 0$, hence there are two real roots.

Note: If the discriminant is a perfect square, the two roots are rational numbers. If the discriminant is not a perfect square, the two roots are irrational numbers containing a radical.

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