What you need to know

Things to remember:

• We can find the value(s) that satisfy an equation by rearranging to get the variable by itself.

We have seen linear equations before, they’re equations that can be drawn as a straight line (they don’t have powers). So, they usually look something like this:

$$y=5x+2$$

$$y=7-3x$$

$$8y+5x=3$$

A nonlinear equation, however, cannot be drawn as a straight line. These equations involve powers and multiplying variables together. They look something like this:

$$y=x^2+4$$

$$y=\sqrt{x}$$

$$xy=8$$

We’re going to look at two types of questions here, and each will take two steps.

Solve the following nonlinear equation $x^2+4=29$.

Step 1: Get the variable by itself.

Hint: Remember, if we see an addition we subtract it, and if we see plus we subtract it.

$$x^2+4=29$$

$$x^2+4-4=29-4$$

$$x^2=25$$

Step 2: Do the opposite of what you’re doing to the $x$.

Hint: What is the opposite of square? Square rooting!

$$x^2=25$$

$$\sqrt{x^2}=\sqrt{25}$$

$$x=5$$

Note: A square root provides two answers, a positive and a negative ($-5\times-5=25$.

Solve the following nonlinear equation $\sqrt{x}-3=9$.

Step 1: Get the variable by itself.

Hint: Remember, if we see an addition we subtract it, and if we see plus we subtract it.

$$\sqrt{x}-3=9$$

$$\sqrt{x}+3=9+3$$

$$\sqrt{x}=12$$

Step 2: Do the opposite of what you’re doing to the $x$.

Hint: What is the opposite of square rooting? Squaring!

$$\sqrt{x}=12$$

$$\sqrt{x}^2=12^2$$

$$x=144$$

Example Questions

Question 1: Solve the following nonlinear equation $x^2-3=13$.

Step 1: Get the variable by itself.

Hint: Remember, if we see an addition we subtract it, and if we see plus we subtract it.

$$x^2-3=13$$

$$x^2+3=13 +3$$

$$x^2=16$$

Step 2: Do the opposite of what you’re doing to the $x$.

Hint: What is the opposite of square? Square rooting!

$$x^2=16$$

$$\sqrt{x^2}=\sqrt{16}$$

$$x=4$$

Note: A square root provides two answers, a positive and a negative ($-4\times-4=16$.)

Question 2: Solve the following nonlinear equation $\sqrt{x}+8=28$.

Step 1: Get the variable by itself.

Hint: Remember, if we see an addition we subtract it, and if we see plus we subtract it.

$$\sqrt{x}+8=28$$

$$\sqrt{x}+8-8=28-8$$

$$\sqrt{x}=20$$

Step 2: Do the opposite of what you’re doing to the $x$.

Hint: What is the opposite of square rooting? Squaring!

$$\sqrt{x}=20$$

$$\sqrt{x}^2=20^2$$

$$x=400$$