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

Answer 1

#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.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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