College Algebra : Radical Equations

Study concepts, example questions & explanations for College Algebra

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Example Questions

Example Question #1 : Radical Equations

Solve for 

 

 

 

 

Possible Answers:

There are no real solutions. 

Correct answer:

There are no real solutions. 

Explanation:

Solve for 

 

 

First isolate one of the radicals; the easiest would be the one with more than one term. 

 

 

Square both sides of the equation, 

 

Expand the right side, 

 

Now collect terms and isolate the remaining radical expression; note the the 's on the left are right sides cancel. 

 

Square both sides, 

 

CHECK THE SOLUTION

We have done all of the algebra correctly, but we can still end up with an erroneous solution due to the squaring operation (a very similar problem arises when dealing with absolute value equations). Once you arrive at a solution, make sure you check that the solution works. If it does not work, and you know your algebra was right, then there are no real solutions.

 

 

Therefore there are no real solutions. 

 

 

Example Question #2 : Radical Equations

Solve for 

 

 

 

Possible Answers:

The are no solutions. 

Correct answer:

Explanation:

Solve for 

 

 

The first step for equations with radicals is always to isolate the radical onto one side of the equation, 

 

Square both sides, 

 

Expand the right side, 

 

If we collect all terms to one side of the equation we will end up with a quadratic equation.

(Notice that 3*(-4) is -12 and 3+(-4) =-1. This is a pattern you should look for when trying to factor quadratics. It does not always work that you can find two numbers such that their product is equal to the constant term and their sum is the coefficient of the linear, or "x" term. In such cases you must use the quadratic formula or complete the square). 

 

Factor the quadratic, 

The solutions for this quadratic are: 

 

CHECK THE SOLUTIONS

One common mistake for students is to assume that these solutions are infact solutions to the original equation. Whenever you work with absolute value equations, or radical equations, you must check the solution carefully to make sure the solutuion actually works. The squaring operation can produce erroneous solutions, and radical equations can often have no solutions at all. As you will see, only one of the two solutions above work in this case.

 

We started with, 

 

Now try , you'll see that the left side will not reduce to the right side: 

 

So  is not a solution.  

 

 

Try 

 

So  is a solution. 

 

 

 

 

 

 

Example Question #3 : Radical Equations

Solve for x:

Possible Answers:

no real solutions

Correct answer:

Explanation:

using the quadratic formula we get

so the possible solutions are  and . However,  is not an actual solution because it is negative and the equation  can only be satisfied by a positive value.

Example Question #4 : Radical Equations

Solve the radical equation:  

Possible Answers:

Correct answer:

Explanation:

Square both sides to eliminate the radical.

Divide by three on both sides to isolate the x.

The answer is:  

Example Question #5 : Radical Equations

Solve the equation:  

Possible Answers:

Correct answer:

Explanation:

Square both sides.

Add 2 on both sides.

Divide by 3 on both sides.

The answer is:  

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