Homework Help: Math: Algebra: Quadratic Formula

The quadratic formula gives us an alternative to Completing the Square when we cannot factor an equation. People often find the Quadratic Formula method easier and more convenient because it does not require many operations on the equation being solved.

For solving an equation in the variable x, the Quadratic Formula is:

To find the solutions to an equation, we simply need to identify what a, b, and c are, then substitute them into this formula, and simplify.

First Example (Two Solutions)

We begin applying the Quadratic Formula by putting the equation in the following form:

Where a, b, and c are constants

This means that each term in the equation must be on the left side, just like when we are factoring or Completing the Square. So we subtract from each side.

Now for consistency, we will rearrange the terms so that they are in the same order: The x² term first, the x term second, and the constant term last.

Now by comparing our equation with "ax² + bx + c = 0", we can see that a must equal 1, b must equal 1, and c must equal .

Now that the values of a, b, and c have been determined, we may return to the quadratic formula and use substitution. (Remember to use parentheses when substituting to avoid problems with negative signs.)

We must now simplify this equation keeping the Order of Operations in mind. We begin by simplifying (1)².

Next, we simplify multiplication. We see that 11 is equal to 15:

Now 1 and 15 are added, resulting in 16.

The square root of 16 is 4.

You may recall that the methods of solving by factoring and solving by completing the square required you to split each problem into multiple subproblems to obtain multiple solutions. Since we are again looking for more than one solution, we must split this problem in two.

So far, we have been carrying the ± sign through the problem. Now, we will create two problems, one with a plus sign, and one with a minus sign.

and

Simplifying the first subproblem gives

Simplifying the second subproblem gives

We can now combine these two solutions into the solution to the original example problem:

Second Example: One Repeated Solution

Examine the following problem.

We are using decimal numbers in place of fractions here, but this is a matter of choice. Using decimal numbers will not require us to use the Quadratic Formula any differently.

In this case, the equation is already in the correct format. Using "ax² + bx + c = 0," we must now determine a, b, and c.

Now we can substitute into the Quadratic Formula:

Now we begin simplifying by replacing (22)² with 484.

Simplifying multiplication, we see that -4(10)(12.1) is equal to -484, and 2(10) is equal to 20.

Notice that we cannot create two different subproblems because +0 and -0 are the same value. Therefore, we simply drop the ±0 from the problem.

Without the two subproblems, we will only have one solution.

Imaginary and Complex Solutions

Consider the following problem.

We can identify that

Now, substituting this into the Quadratic Formula:

Simplifying the expression under the radical gives:

Notice that there is a negative number under the square root symbol. If you are familiar with imaginary numbers, you know that the square root of a negative number is an imaginary number, which will cause the solutions to this example to also be imaginary or complex numbers.

In this lesson, we chose to expose you to this situation, but not to provide details on imaginary and complex solutions. We will add a lesson on this topic after we have introduced imaginary and complex numbers.

Homework Help: Math: Algebra

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