You have studied how to complete the square to write the equation of a parabola in vertex form. When a quadratic equation is in vertex form, you can easily find the vertex. You can also set the equation equal to zero and solve to find the <EQUATION> intercepts. Answer the following questions. Click “Submit” when finished.

Consider the general form of a quadratic equation <EQUATION>. Let's solve this quadratic equation for <EQUATION> by completing the square. We start by factoring <EQUATION> out of the first two terms. We complete the square by adding <EQUATION> and <EQUATION>. The first three terms inside the braces are a perfect square trinomial, so the three terms can be replaced with \n <EQUATION>. Now we can distribute the <EQUATION> to remove the curly brackets. Now notice that <EQUATION> occurs only in the square brackets. We can begin to isolate the squared quantity by subtracting <EQUATION> and adding <EQUATION> to each side. Then, we multiply each side by <EQUATION>.

Now that we have isolated the squared quantity, we can take square roots to give us two equations. We can solve for <EQUATION> by subtracting <EQUATION> from each side. We can simplify the expression under the square root sign. Let's start by squaring the fraction. We can then find a common denominator and combine the two terms under the square root sign. We add fractions with a common denominator by adding numerators. We take the square root of a fraction by taking the square root of the numerator and the square root of the denominator. The square root of the denominator, <EQUATION>, is <EQUATION>.

The two new fractions already have the same common denominator, <EQUATION>, so combine them. We can use the symbol <EQUATION> to indicate two cases, the <EQUATION> and the <EQUATION>. This allows us to combine the two solutions into one formula, called the |B| quadratic formula |/B| . This formula can be used to quickly solve for x in any quadratic equation of the form <EQUATION>.

Lets use the quadratic formula to solve <EQUATION>. To use the quadratic formula, we need to determine the values of <EQUATION>, <EQUATION>, and <EQUATION>. We can see that <EQUATION>, because this is the coefficient of <EQUATION>. <EQUATION>, because this is the coefficient of <EQUATION>. And the constant term, <EQUATION>, is <EQUATION>.

Now we can substitute these three values into the quadratic formula. We can begin to evaluate the quadratic formula by multiplying. Then subtract <EQUATION> from <EQUATION> so that we can evaluate the square root. The square root of <EQUATION> is <EQUATION>. Now we break the equation into the <EQUATION> case and the <EQUATION>, and we evaluate each. The two solutions are <EQUATION> and <EQUATION>.

Use the quadratic formula to solve the equation. Click “Submit” when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education