Solving Quadratic Equations Using the Quadratic Formula
The quadratic formula gives us a powerful tool for solving equations. It is powerful because it can be used to find the solutions to any quadratic equation. Here’s how it works.
Let us assume that we have a quadratic equation in the form:
Note that the letters a, b and c will be numbers in specific examples. To get a quadratic equation into the above form It may be necessary to rearrange the terms, but once it looks like this, we are ready to apply the quadratic formula.
The quadratic formula states that the solutions of the above equation are:
This solution comes from solving the equation by completing the square.
Since we have a sign, we have to be sure that we calculate x using “+” and then calculate x again using “–“. This is what gives us the two solutions that every quadratic equation has.
Let’s try solving one of the equations that was solved in the tutorial on completing the square and make sure that we get the same solution using the quadratic formula.
The first step is to rearrange terms so that the equation is in the correct form (i.e. ) for using the quadratic formula. One side needs to equal to zero, so we subtract 3 from both sides and get:
Now we identify the values of a, b and c. The coefficient of is 1, so . The coefficient of x is 6, so . The constant term in this equation is -7, so .
Plugging these values into the quadratic formula gives us:
Let’s simplify under the radical first, so we get
The square root of 64 is 8 and remembering that the sign means we really have two values for x, we get the following two solutions:
These are exactly the same solutions we got by completing the square so the formula checks out.
Now let’s try one that leads to something rather strange. Let’s use the quadratic formula to solve the equation:
It’s already in the proper form for using the quadratic equation, so we do not need to rearrange any terms. In this case, , and . Now when we plug these numbers into the quadratic formula we get this:
This simplifies to:
There’s a problem here, though. The square root of a negative number is not a real number. You may think that this means that the quadratic equation has no solution, but that is not quite right. This quadratic has no “real” solutions, but it does have what are called complex solutions. Graphically speaking, if a quadratic equation has no real solutions, then its graph does not pass through the x-axis. This concept is addressed further in the tutorial entitled Finding Points of Intersection.