Quadratics

The Quadratic Formula

The Quadratic Formula

Trinomials are not always easy to factor. In fact, some trinomials cannot be factored. Thus, we need a different way to solve quadratic equations. Herein lies the importance of the quadratic formula:

Given a quadratic equation ax ² + bx + c = 0 , the solutions are given by the equation

x =

Example 1: Solve for x : x ² + 8x + 15.75 = 0

a = 1 , b = 8 , and c = 15.75 .
x =

=
=
=
= or
= - or -

Thus, x = - or x = - .

Example 2: Solve for x : 3x ² - 10x - 25 = 0 .

a = 3 , b = - 10 , and c = - 25 .
x =

=
=
=
=
= or
= 5 or -

Thus, x = 5 or x = - .

Example 3: Solve for x : -3x ² - 24x - 48 = 0 .

a = - 3 , b = - 24 , and c = - 48 .
x =

=
=
=
=
= = - 4

Thus, x = - 4 .

Example 4: Solve for x : 2x ² - 4x + 7 .

a = 2 , b = - 4 , and c = 7 .
x =

=
=
=

Since we cannot take the square root of a negative number, there are no solutions. (The graph of this quadratic polynomial will therefore be a parabola that never touches the x -axis.)

The Discriminant

As we have seen, there can be 0 , 1 , or 2 solutions to a quadratic equation, depending on whether the expression inside the square root sign, (b ² - 4ac) , is positive, negative, or zero. This expression has a special name: the discriminant.

If the discriminant is positive--if b ² -4ac > 0 --then the quadratic equation has two solutions.
If the discriminant is zero--if b ² - 4ac = 0 --then the quadratic equation has one solution.
If the discriminant is negative--if b ² -4ac < 0 --then the quadratic equation has no solutions.

Example: How many solutions does the quadratic equation 2x ² + 5x + 2 have?

a = 2 , b = 5 , and c = 2 .
b ² -4ac = 5² -4(2)(2) = 25 - 16 = 9 > 0 .

Thus, the quadratic equation has 2 solutions.