# Roots and Radicals

## Solve Equations with Square Roots


### Solve Radical Equations

In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.
As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will square both sides in order to remove the radical sign and solve for the variable inside.
But remember that when we write  we mean the principal square root. So  always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make  negative. This algebraic solution would not be a solution to the original radical equation; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too.
Now we will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring.
When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.
If one side of the equation is a binomial, we use the binomial squares formula when we square it.
Don’t forget the middle term!
When there is a coefficient in front of the radical, we must square it, too.
Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both sides of the equation again.

### Use Square Roots in Applications

As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.
We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give us a plan for solving applications with formulas from any discipline.
We used the formula  to find the area of a rectangle with length L and width W. A square is a rectangle in which the length and width are equal. If we let s be the length of a side of a square, the area of the square is .
The formula  gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area? Then we need to solve the equation for s.

We can use the formula  to find the length of a side of a square for a given area.
We will show an example of this in the next example.
Another application of square roots has to do with gravity.
For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting  into the formula.

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.
Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.

### Key Concepts

1. To Solve a Radical Equation:


2. Solving Applications with Formulas


3. Area of a Square



4. Falling Objects


5. Skid Marks and Speed of a Car



### Practice Makes Perfect

Solve Radical Equations
In the following exercises, check whether the given values are solutions.
In the following exercises, solve.
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?