The Quadratic Formula

amath The Quadratic Formula defines a way of finding the roots of quadratic equations, especially those that are not factorable. Given a quadratic equation, a x^2+b x+c=0, where $a\neq0$, we can complete the square to derive the quadratic formula.
First, we divide through by a:
x^2+b/a x=-c/a
Then, we complete the square:
x^2+b/a x+b^2/(4 a^2)=-c/a+b^2/(4 a^2)
We can re-arrange the two sides:
(x+b/(2 a))^2=(b^2-4a c)/(4 a^2)
Take the square root of both sides:
|x+b/(2 a)|=sqrt(b^2- 4a c)/(2 a)
Finally, we subtract b/(2 a) from both sides:
x_(1,2)=(-b+-sqrt(b^2-4a c))/(2a)

This is the quadratic formula:

x_(1,2)=(-b+-sqrt(b^2-4a c))/(2a)

Please use the quadratic formula to solve the following examples:

Example 1: Find the roots of 3 x^2-2x-1=0.

x_(1,2)=(-b+-sqrt(b^2-4a c))/(2a)=(-(-2)+-sqrt((-2)^2-4(3)(-1)))/(2(3))=(2+-sqrt(16))/6
x_1=-1/3 or x_2=1

Example 2: Find the roots of 3 x^2-2x+1=0.

x_(1,2)=(-b+-sqrt(b^2-4a c))/(2a)=(-(-2)+-sqrt((-2)^2-4(3)(1)))/(2(3))=(2+-sqrt(-8))/6=(1+-isqrt(2))/3

The expression inside the square root of the quadratic formula, b^2-4a c, is called the discriminant and labeled by $\Delta$.
Keeping in mind that the real roots of a quadratic equation, a x^2+b x+c=0 are the x-intercepts of the parabola, y=a x^2+b x+c, the next page illustrates the importance of the discriminant in determining the number of real roots of a quadratic equation. endamath