Algebra Know-It-ALL

381

CHAPTER

23 Quadratic Equations with Complex Roots

Now that we know a little bit about what to expect when the discriminant of a quadratic equa-

tion is negative, let’s explore this territory in more detail. A negative discriminant indicates

that the roots of a quadratic are imaginary or complex.

Complex Roots by Formula

The quadratic formula, like basic addition facts and multiplication tables, is worth commit-

ting to memory. (That’s why I keep repeating it. Here it is again!) When we have a quadratic

equation of the form

ax^2 +bx+c= 0

then the roots can be found using the formula

x= [−b± (b^2 − 4 ac)1/2] / (2a)

Square root of the discriminant

The discriminant of a quadratic equation is equal to the square of the coefficient of x, minus

4 times the product of the coefficient of x^2 and the stand-alone constant. In the quadratic

formula as stated above, the discriminant d is

d=b^2 − 4 ac

When the coefficients and constant in a quadratic are real numbers, then the discriminant

is always a real number. If d> 0, then the square root of d can be either a positive real or its

additive inverse. If d= 0, then the square root of d is 0. If d< 0, then the square root of d can

be either of two values, one positive imaginary and the other negative imaginary.

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