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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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