How to Solve x^2 - 6x = 16

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will go over the steps to solve the equation x^2 - 6x = 16. We will examine two different solving processes, and learn how to check our work for accuracy.

Steps to Solve

We will look at two methods of solving x 2 - 6x = 16. The first uses factoring, and the second uses the quadratic formula.

Solving Using Factoring

When solving by factoring, we will use the following steps:

1.) Get all non-zero terms on one side of the equation, and zero on the other.

2.) Factor the non-zero side of the equation.

3.) Set each factor equal to zero and solve.

The first thing we want to do is get all non-zero terms on one side of the equation. To do this, we subtract 16 from both sides of the equation as shown.


solve1


The next step is to factor the non-zero side of the equation, so we want to factor x 2 - 6x - 16. When the coefficient in front of x 2 is one, as it is in our case, we can use the process shown in the image to factor.


How to Factor
solve2


We want to factor x 2 - 6x - 16, so we want to fill the blanks of (x + _____)(x + _____) with two numbers that when multiplied, are equal to -16 and when added, are equal to -6. We can find this by listing the numbers that multiply together to equal -16, and then check each one to see if they add up to -6.

Factors of 16 Sum
1 and -16 1 + (-16) = -15
-1 and 16 -1 + 16 = 15
2 and -8 2 + (-8) = -6
-2 and 8 -2 + 8 = 6
4 and -4 4 + (-4) = 0

The only pair of factors of 16 that add up to -6 are 2 and -8, so these are the numbers we will use to fill in the blanks. That is, x 2 - 6x - 16 = (x + 2)(x - 8).

The last step is to set each of these factors equal to zero and solve as shown.


solve3


We see that x = -2 or x = 8.

Solving Using the Quadratic Formula

Another way to solve this problem is using the quadratic formula. The quadratic formula gives solutions to an equation of the form ax 2 + bx + c = 0.


The Quadratic Formula
solve4


To solve using the quadratic formula, we use the following steps:

1.) Put the equation in the form ax 2 + bx + c = 0 by getting all non-zero terms on one side and zero on the other side.

2.) Identify the values of a, b, and c in your equation from step 1.

3.) Plug your values for a, b, and c into the quadratic formula and simplify.

In our example, we want to solve x 2 - 6x = 16. The first step is the same as the first step when we used factoring. We get all the non-zero terms on one side by subtracting 16 from both sides to get x 2 - 6x - 16 = 0.

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