### Learning Outcomes

- Solve linear equations
- Use the zero product principle to solve quadratic equations that can be factored
- Use the square root property to solve quadratic equations
- Write function equations using given conditions

The Separable Equations section will require that you set factors of various degrees equal to zero and solve the resulting equation. We will also be asked to find a function equation using given conditions. Here we review all of these skills.

## Solve Linear Equations

A linear equation is an equation whose highest power on the variable is one. If the equation is in the form [latex]ax+b=c[/latex], where *x* is the variable, to solve the equation as before, first “undo” the addition and subtraction and then “undo” the multiplication and division.

### Example: Solving a Linear Equation

Solve [latex]3y+2=11[/latex].

### Example: Solving a Linear Equation

Solve [latex]3x+5x+4-x+7=88[/latex].

Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[/latex].

To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It does not matter which term gets moved, [latex]4x[/latex] or [latex]2x[/latex]; however, to avoid negative coefficients, you can move the smaller term.

### Example: Solving a Linear Equation

Solve: [latex]4x-6=2x+10[/latex]

### Try It

Solve the linear equation [latex]7x-15=3x+4[/latex].

## Solve Quadratic Equations

Often, the easiest method of solving a quadratic equation is by **factoring**.

### How To: Given a quadratic equation, Solve it by factoring

- Set the quadratic equation equal to 0.
- Factor.
- Set each factor equal to 0 and solve for the variable.

### Example: Solving A Quadratic Equation by Factoring

Factor and solve the equation: [latex]{x}^{2}+x - 6=0[/latex].

### Example: Solving A Quadratic Equation by Factoring

Factor and solve the equation: [latex]{8x}^{2}+2x - 3=0[/latex].

### Try It

When there is no linear term in the equation, another method of solving a quadratic equation is by using the **square root property**, in which we isolate the [latex]{x}^{2}[/latex] term and take the square root of the number on the other side of the equal sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the [latex]{x}^{2}[/latex] term so that the square root property can be used.

### Example: Solving a Quadratic Equation Using the Square Root Property

Solve the quadratic using the square root property: [latex]{x}^{2}=8[/latex].

### Example: Solving a Quadratic Equation Using the Square Root Property

Solve the quadratic equation: [latex]4{x}^{2}+1=7[/latex]

### Try It

Solve the quadratic equation using the square root property: [latex]3{\left(x - 4\right)}^{2}=15[/latex].

### Try It

## Write Function Equations Using Given Conditions

*(See Module 4, Skills Review for Basics of Differential Equations)*