How do you find the value of the discriminant and determine the nature of the roots $4x² – 8x = 3$ ?

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Answer 1 · 2018-02-09T21:11:18

$Delta = 112 > 0$ is not a perfect square, so this quadratic equation has two distinct real but irrational roots.

Given:

$4x^2-8x=3$

Subtract $3$ from both sides to get:

$4x^2-8x-3 = 0$

This is in the standard form $ax^2+bx+c = 0$ , with $a=4$ , $b=-8$ and $c=-3$ .

It has discriminant $Delta$ given by the formula:

$Delta = b^2-4ac = (-8)^2-4(4)(-3) = 64+48 = 112$

Since $Delta > 0$ this quadratic has two distinct real roots.

Note however that $Delta = 112$ is not a perfect square. Hence we can deduce that the roots are irrational.

In general, we find:

If $Delta > 0$ is a perfect square, then the quadratic equation has two distinct rational roots.
If $Delta > 0$ is not a perfect square, then the quadratic equation has two distinct real, but irrational roots.
If $Delta = 0$ then the quadratic equation has one repeated rational real root.
If $Delta < 0$ then the quadratic equation has no real roots. It has a complex conjugate pair of non-real roots. If $-Delta$ is a perfect square then the imaginary coefficient is rational.

How do you find the value of the discriminant and determine the nature of the roots 4x² – 8x = 3 4x² – 8x = 3 ?