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When working with some topics such as kinetic energy or the free-fall of an object, radical equations and inequalities can be used to find desired values. This lesson will cover this type of equation and inequality in detail and to solve them.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

High School Mountain Bike Race

LaShay and Ali are members of the Jefferson High School Mountain Bike Team. They are both training hard for the upcoming Mountain Bike Race.

High School Mountain Bike Race
The following equations represent their paces.
In these equations, and represent the distance traveled by each person, in miles, after hours. After how many hours will they both cycle the same distance?
Discussion

Radical Equation

A radical equation is an equation that has an independent variable in the radicand of a radical expression, or an independent variable with a rational exponent. A radical equation with a root that has an index of is called a square root equation. If the index of the root is the equation is a cube root equation.

Variable in a Radicand Variable With a Rational Exponent
Discussion

Solving Radical Equations Algebraically and the Concept of Extraneous Solutions

Radical equations can be solved algebraically by using inverse operations and the Properties of Equality. Because some roots can only take certain values, this process can produce solutions that do not actually satisfy the equation. This type of solutions have a special name.

Concept

Extraneous Solution

Solving equations by using inverse operations may lead to solutions that do not satisfy the original equation. These solutions are called extraneous solutions, and they occur primarily in radical equations when the radical is eliminated. Below, an example radical equation is shown.
By using inverse operations and Properties of Equality, the solutions to this equation are found to be and However, they produce different outcomes when substituted into the original equation.
Substitute Simplify
For the example equation, is a valid solution but does not result in a true statement. Therefore, it is an extraneous solution.
Method

Solving a Radical Equation Algebraically

Consider an example radical equation.
Four steps must be followed to solve this equation.
1
Isolate the Radical
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When solving a radical equation, it is necessary to isolate the radical expression on one side. Use inverse operations to achieve this.
2
Eliminate the Radical
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The radical can be undone or eliminated by raising it to the same power as its index. Here, the radical is a square root, so the index is Therefore, both sides of the equation must be raised to the power of
3
Solve the Equation
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Once the radical has been eliminated, the resulting equation can be solved.
Now the equation can be solved for the variable. In this case, a quadratic equation was obtained and must be solved as such. First, it will be rearranged, then the Quadratic Formula will be used.
Solve using the quadratic formula

Add and subtract terms

Calculate quotient

The solutions to the equation are and
4
Check for Extraneous Solutions
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The solutions found in the previous step might be extraneous solutions. Therefore, they must be verified in the original equation. First, the solution is tested.
Evaluate right-hand side
Since does not satisfy the radical equation, it is an extraneous solution. Next, can be checked in the same way.
Evaluate right-hand side
Since makes a true statement, it is a solution to the radical equation. The given equation can be said to have one solution and one extraneous solution.
Example

Solving Simple Radical Equations

The rectangular board in LaShay's class is wide feet by feet long.

Radical Equation on the Black Board
Knowing that the perimeter of the board is feet, LaShay wants to write an equation that represents the perimeter.
Next, she wants to use the equation to find the value of

Hint

The perimeter of a rectangle is where is the length and is the width of the rectangle.

Solution

The perimeter of a rectangle is where is the length and is the width of the rectangle. A radical equation representing the perimeter of the board can be written by substituting and into the formula.
From here, the first step to solving this equation is to isolate the radical expression.
Now that the radical expression is isolated, the radical equation can be solved by eliminating the radical.
Finally, will be substituted into the original equation to see whether it is an extraneous solution.
Evaluate right-hand side
Since it satisfies the original equation, is a valid solution.
Example

Practicing Radical Equations in an Amusement Park

After an exhausting school day, LaShay went to an amusement park to have fun with her friends. The first thing they did was ride on a Ferris wheel.

Ferris-wheel.jpg

Everything was going well until LaShay accidentally dropped her phone from the top of the Ferris wheel. The following radical equation models the time passed after the phone was dropped.
Here, is the time in seconds after the phone was dropped, is the height of the phone above the ground in meters, and is LaShay's height in meters above the ground at the moment she dropped the phone. If LaShay was meters high when she dropped the phone, how many meters above the ground would the phone be after seconds?

Hint

Substitute the given values into the equation and solve for

Solution

To find how many meters above the ground the phone would be after seconds, begin by substituting and into the formula.
Next, the radical equation can be solved algebraically to find the value of
Finally, check if the solution satisfies the original equation.
Evaluate right-hand side

\AddTerm

Therefore, LaShay's phone would be meters above the ground seconds after she dropped it. Sadly, it is very unlikely that the phone will survive the fall.
Example

Radical Equations With Variables Outside the Radical

After an exciting day at the amusement park, even if the math did not help LaShay save her phone from smashing against the ground, her interest in radical equations increased. Right now, LaShay and Ali are trying to solve the following radical equation.

Radical Equation on the Black Board

After some work, they managed to solve the equation. However, they came up with different solutions.

Radical Equation on the Black Board
Solve the given equation and decide who is correct. If LaShay's solution is wrong, she will need to study more.

Hint

Start by isolating the radical expression.

Solution

When solving a radical equation, the first thing to do is isolate the radical expression.
Then, raise both sides of the equation to the second power in order to get rid of the root.
Rewrite
Since the obtained equation is a quadratic equation, the Quadratic Formula will be used to solve it.
Evaluate right-hand side

Add and subtract terms

Calculate quotient

Finally, check for extraneous solutions.
Substitute Simplify

Note that solving the equation by using inverse operations and Properties of Equality produced two solutions, and However, does not satisfy the original equation. Therefore, it is an extraneous solution. Conversely, since satisfies the original equation, it is a solution. As a result, only Ali is correct and LaShay is incorrect.

Discussion

Solving Radical Equations Graphically

Radical equations can be solved algebraically and graphically. Solving them algebraically sometimes produces extraneous solutions, whereas solving them graphically does not produce extraneous solutions. However, while algebraic solutions are generally exact, graphical solutions are often approximated. Consider an example radical equation.
To solve this equation graphically, three steps must be followed.
1
Write Two Functions
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Start by writing two functions. Each side of the equation represents a function.
The first is a radical function and the second is a linear function.
2
Graph Both Functions on the Same Coordinate Plane
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To graph the functions, make a table of values for each. Remember that the radicand cannot be negative!

Now the points for each function found in the table will be plotted on the same coordinate plane. Then the points of the linear function will be connected with a straight line, and the points of the radical function will be connected with a smooth curve.

Linear and Radical Functions


3
Identify the Point of Intersection
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The coordinate of the point of intersection of the functions gives the solution of the original equation.

Linear and Radical Functions

The curve and the line intersect at This means that the solution to the equation is

Example

Completing Homework by Solving Radical Equations Graphically

After seeing LaShay's hard work, her teacher gives her extra credit homework so that she can have more practice with radical equations.

Radical Equations With Radical Expressions on Both Sides
LaShay knows that if she solves this equation graphically, she will not have any extraneous solutions. Solve the radical equation graphically and find the value of

Hint

Create two functions, one for each side of the equation.

Solution

To solve a radical equation graphically, both sides of the equation need to be represented by functions.
Now that there are two radical functions, they can be graphed by making a table of values. Since the radicands cannot be negative, the domain of each function will be found.
Therefore, the table of values for the first function will be made for values of greater than or equal to For the second function, the values must be greater than or equal to
- -

Now the points found in both functions can be plotted in the same coordinate plane. Then they can be connected with smooth curves.

Graph of Two Radical Functions

Finally, the coordinate of the point of intersection of the curves gives the solution of the original equation.

Graph of Two Radical Functions

Because the point of intersection is at the solution to the equation is

Discussion

Radical Inequalities

A radical inequality is an inequality that has an independent variable in a radicand or an independent variable with a rational exponent.

Variable in a Radicand Variable With a Rational Exponent

This type of inequality can be solved algebraically.

Method

Solving Radical Inequalities Algebraically

Radical inequalities can be solved in a similar way to radical equations. The difference is that here, if the index is even, the values that make the radicand non-negative must be identified first. Consider an example radical inequality.
To solve this inequality, five steps must be followed.
1
Identify the Values that Make the Radicand Non-negative
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Since the index is even, the radicand cannot be negative.
Solve this inequality for
For this inequality, the values of must always be greater than or equal to
2
Isolate the Radical
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Next, isolate the radical by using inverse operations.
3
Eliminate the Radical Symbol
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Since the index of the root is the radical can be undone or eliminated by raising both sides of the inequality to the second power.
4
Solve the Inequality
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When the radical has been eliminated, the resulting inequality can be solved.
The values of that satisfy the given inequality are less than or equal to Recall that at the beginning it was found that must be greater than or equal to Therefore, the solution set of the inequality can be written as a compound inequality.
5
Test Values to Check the Solution
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Finally, some values can be tested to confirm the solution. In this case, three test values will be used. The first value will be less than the second greater than or equal to and less than or equal to and the third value will be greater than In this case, and will be tested.

Substitute Simplify