Chapter Fourteen — Quadratic Equations and Inequalities
The Humongous Book of Algebra Problems
A positive discriminant indicates that the equation has two unique real roots.
In other words, there are two different, real number solutions to the quadratic.
The solutions may be rational (which means that the equation can be solved
by factoring) or irrational (which means that the equation must be solved by
completing the square or the quadratic equation), but both solutions are real
A zero discriminant is an indication of a double root. The quadratic still has
two solutions, but the solutions are equal. For instance, the quadratic equation
(x + 5)(x + 5) = 0 has two roots: x = –5 and x = –5. Because a single real root is
repeated, x = –5 is described as a double root.
If a quadratic equation has a negative discriminant, there are no real number
solutions. That does not mean the equation has no solutions, but instead that
the solutions are complex numbers.
14.30 Given the equation 3x
+ 7x + 1 = 0, use the discriminant to predict how many
of the roots (if any) are real numbers, then calculate the roots to verify the
prediction made by the discriminant.
The equation has form ax
+ bx + c = 0, so substitute a = 3, b = 7, and c = 1 into
the discriminant formula.
According to Problem 14.29, the equation 3x
+ 7x + 1 = 0 has two positive
real roots because the discriminant is positive (37 > 0). Calculate the roots by
applying the quadratic formula.
The solution to the equation is or .
In case you
it, the discriminant
– 4ac is the part
of the quadratic
formula that’s inside
the radical. When you
plug everything into the
there’s no need to
again—just plug in
the discriminant, 37.