# What is the discriminant of 4x^2-64x+145=-8x-3 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.

$D = 48$

#### 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;

4x^2−64x+145=−8x−3

First transform it to standard form of the quadratic equation.

4x^2−64x+145+8x+3=0 $\implies$ Added $8 x$ and $3$ on both side.
or, $4 {x}^{2} - 56 x + 148 = 0 \implies$ Combined like terms.
or, ${x}^{2} - 14 x + 37 = 0 \implies$ Divided both side by 4.

Now compare the above equation with quadratic equation $a {x}^{2} + b x + c = 0$, we get $a = 1 , b = - 14 \mathmr{and} c = 37$.

Hence the discriminant (D) is given by;

$D = {b}^{2} - 4 a c$
$\implies D = {\left(- 14\right)}^{2} - 4 \cdot 1 \cdot 37$
$\implies D = 196 - 148$
$\implies D = 48$

Therefore the discriminant of a given equation is 48.

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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