If the discriminant b2 - 4ac equates to zero, the radical in the quadratic formula becomes zero.

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In this case the roots are equal; such roots are periodically called double roots.

Consider the equation

9x2 + 12x + 4 = 0

Comparing via the basic quadratic, we notice that

a = 9, b = 12, and c = 4

The discriminant is


Therefore, the roots are equal.

CHECK: From the formula


The ehigh quality of the roots is for this reason proved.

The roots deserve to be equal only if the trinomial is a. perfect square. Its components are equal. Factoring the trinomial in

9x2 + 12x + 4 = 0

we view that

(3x + 2)2 = 0

Due to the fact that the variable 3x + 2 is squared, we actually have

3x + 2 = 0

twice, and we have



The reality that the very same root need to be counted twice defines the usage of the term "double root." A double root of a quadratic equation is constantly rational bereason a twin root deserve to take place only when the radical vanishes.

REAL AND UNEQUAL ROOTS When the discriminant is positive, the roots have to be genuine. Also they should be unequal considering that equal roots take place just as soon as the discriminant is zero.

Rational Roots .

If the discriminant is a perfect square, the roots are rational. For example, consider the equation

3x2 - x - 2 = 0

in which

a = 3, b = -1, and c = -2

The discriminant is


We see that the discriminant, 25, is a perfect square. The perfect square indicates that the radical in the quadratic formula deserve to be rerelocated, that the roots of the equation are rational, and that the trinomial can be factored. In various other words, once we evaluate the discriminant and also discover it to be a perfect square, we recognize that the trinomial deserve to be factored.

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from which


We check out that the indevelopment derived from the discriminant is correct. The roots are actual, unequal, and also rational.