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:

Example 1: Solve for x: x² + 8x + 15.75 = 0

a = 1, b = 8, and c = 15.75.
x =
Thus, x = - or x = - .

Example 2: Solve for x: 3x² - 10x - 25 = 0.

a = 3, b = - 10, and c = - 25.
x =
Thus, x = 5 or x = - .

Example 3: Solve for x: -3x² - 24x - 48 = 0.

a = - 3, b = - 24, and c = - 48.
x =
Thus, x = - 4.

Example 4: Solve for x: 2x² - 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.)

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, (b² - 4ac), is positive, negative, or zero. This expression has a special name: the discriminant.

If the discriminant is positive--if b² -4ac > 0--then the quadratic equation has two solutions.
If the discriminant is zero--if b² - 4ac = 0--then the quadratic equation has one solution.
If the discriminant is negative--if b² -4ac < 0--then the quadratic equation has no solutions.

Example: How many solutions does the quadratic equation 2x² + 5x + 2 have?

a = 2, b = 5, and c = 2.
b² -4ac = 5² -4(2)(2) = 25 - 16 = 9 > 0.

Thus, the quadratic equation has 2 solutions.