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.