Algebra Know-It-ALL

378 Quadratic Equations with Real Roots

The discriminant

In the second example above, b^2 − 4 ac= 0. In that case, it doesn’t matter whether we add

(b^2 − 4 ac)1/2 to −b or subtract (b^2 − 4 ac)1/2 from –b in the formula. This gives us a quick way

to tell whether a quadratic equation has two roots or only one. The quantity b^2 − 4 ac is called

thediscriminant of the general quadratic equation

ax^2 +bx+c= 0

If the discriminant is a positive real number, then the associated quadratic has two real roots.

If the discriminant is equal to 0, then the quadratic has one real root with multiplicity 2.

There are plenty of quadratic equations in which the discriminant is a negative real num-

ber. Here’s an example:

4 x^2 − 4 x+ 36 = 0

This is almost exactly the same equation as we solved in the second example above. The only

difference is that the second coefficient is 4 rather than 24. The discriminant here is

b^2 − 4 ac= 42 − 4 × 4 × 36

= 16 − 576

=− 560

When we apply the quadratic formula, we must take the square root of −560. That’s an imagi-

nary number! A negative discriminant gives us imaginary or complex roots. We’ll explore

situations like this in the next chapter. We’ll also see what happens when one or more of the

coefficients in a quadratic equation are imaginary or complex.

Are you confused?

Do you wonder how we came up with the constant b^2 /(4a^2 ) to make a perfect square in the process of

deriving the quadratic formula? Again, this is the “sixth sense” at work, a form of intuition that you can

develop only with practice.

Here’s a challenge!

In the derivation of the quadratic formula, we made a “quantum leap” when we claimed that the polynomial

x^2 + (b/a)x+b^2 /(4a^2 )

is a perfect square that can be factored into

[x+b/(2a)]^2

Show that this is actually true.