Solving Quadratic Equations Help

Quadratic equations can be solved in 4 different ways:

Graphing
Factoring
Completing the Square
Quadratic Formula

It does not matter which method you use but you should end up with the same answer.

Factoring is generally the easiest.

The quadratic formula can always be used if you can't find the answer any other way.

The following example illustrates the steps to follow when solving by each method.

Solve: f(x) = x² - 2x - 8

Graphing:

1.  Plot the function.  A graphing calculator
     works well for this. However, setting up
     an x, y table and plotting those points can
     also produce reasonable results.

2. The solutions are the points at which the
graph crosses the x-axis, or where y = 0.

3. The answer for this problem is:
x = -2 and x = 4.

Factoring:

1. Break the function down as the product of
two binomials.

2. Set the factored function equal to zero.

3. Use the Zero Product Property to solve for x.

1. f(x) = x² - 2x - 8 = (x - 4) (x + 2)

2. (x - 4) (x + 2) = 0

3. (x - 4) = 0 x = 4
(x + 2) = 0 x = -2

Completing the Square:

1. Set the function equal to zero.

2. Move the constant term to the right hand
side of the equation.

3.  Square x plus 1/2 of the x coefficient to
     make the left hand side a perfect square
     but remember to also add the square of
     1/2 of the x coefficient to the right hand
     side of the equation to maintain equality.

4. Take the square root of both sides of
the equation.

5. Solve for x.

1. x² - 2x - 8 = 0

2. x² - 2x = 8

3. (x - 1)² = 8 + 1 = 9

4. x - 1 = ± 3

5. x = 1 ± 3 x = 4 and x = -2.

Quadratic Formula:

1. Set the function equal to zero.

2. Identify a, b and c.

Where: ax² + bx + c = 0.

3. Plug a, b and c into the Quadratic Formula:

Note: If the discriminant, b² - 4ac, is:

                positive    2 real roots
                zero         1 real root
                negative    2 imaginary roots

4. Solve for x.

1. x² - 2x - 8 = 0

2. a = 1, b = -2, c = -8

4. x = 1 ± 3 x = 4 and x = -2.