Algebra II: Quadratic Equations

On this page we hope to clear up problems you might be having with quadratic equations. Quadratic equations, or equations of the second degree, such as x^2 + 2x - 5 are probably the most common equation you will see in Algebra II (intermediate algebra). Read on or follow any of the links below to get a better understanding of quadratic equations.

Solving quadratic equations
Quadratic formula
Quadratic form
Quiz on quadratic equations

Solving Quadratic Equations

Any equation of type ax^2 + bx + c = 0 where a, b, and c are constants and a <> 0, is in standard form for a quadratic equation.

Quadratic equations of type ax^2 + bx + c = 0 and ax^2 + bx = 0 (c is 0) can be factored to solve for x.

Example:

1.  Problem: Sovle 3x^2 + x - 2 = 0 for x.
   Solution: Factor.

             (3x - 2)(x + 1)

             Use the principle of zero products, which says, if ab = 0,
             either a, b, or both must be equal to zero.

             3x - 2 = 0,     x + 1 = 0
                 3x = 2,         x = -1
                  x = (2/3)

             x = -1, (2/3)

2.  Problem: Solve 3x^2 + 5x = 0 for x.
   Solution: Factor.

             x(3x + 5) = 0
     
             Use the principle of zero products.

             x = 0,     3x + 5 = 0
                            3x = -5
                             x = -(5/3)

             x = 0, -(5/3)

Quadratic equations of type ax^2 + c = 0 can be solved by solving for x.

Example:

1.  Problem: Solve 3x^2 = 6 for x.
   Solution: Recognize that the equation is quadratic because it is the
             same as 3x^2 - 6 = 0.

             Divide each side by 3.

             x^2 = 2

             Take the square root of each side.

             x = SQRT(2), -(SQRT(2))

Quadratic Formula

Many times you will come across quadratic equations that are not easy to factor or solve. In those cases, there is a special formula called the quadratic formula that you can use to solve any quadratic equation.

The solutions of any quadratic equation, ax^2 + bx + c = 0 is given by the following formula, called the quadratic formula:

                -b ± SQRT(b^2 - 4ac)
            x = --------------------
                        2a

Example:

1.  Problem: Solve 3x^2 + 5x = -1 for x.
   Solution: First find the standard form of the equation and determine
             a, b, and c.

             3x^2 + 5x + 1 = 0
             a = 3
             b = 5
             c = 1

             Plug the values you found for a, b, and c into
             the quadratic formula.

                 -5 ± SQRT(5^2 - 4(3)(1))
             x = -----------------------
                         2 * 3          

             Perform any indicated operations.

                 -5 ± SQRT(25 - 12)
             x = ------------------
                         6

                 -5 ± SQRT(13)
             x = -------------
                       6

             The solutions are as follows:

                 -5 + SQRT(13)  -5 - SQRT(13)
             x = -------------, -------------
                        6             6

Equations in Quadratic Form

Some equations are not quadratic equations, but are in the same form, such as x^4 - 9x^2 + 8 = 0. To solve equations such as that, you make a substitution, solve for the new variable, and then solve for the original variable.

Example:

1.  Problem: Solve x^4 - 9x^2 + 8 = 0 for x.
   Solution: Let u = x^2.  Then substitute u for every
             x^2 in the equation.

             u^2 - 9u + 8 = 0

             Factor.

             (u - 8)(u - 1) = 0

             Utilize the principle of zero products.

             u - 8 = 0,     u - 1 = 0
                 u = 8,         u = 1

             Now substitute x^2 for u and solve the equations.

             x^2 = 8,          x^2 = 1
               x = ±SQRT(8),     x = ± 1
               x = ±2(SQRT(2))

             x = ±2(SQRT(2)), ±1

Take the quiz on quadratic equations. The quiz is very useful for either review or to see if you've really got the topic down.

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