Radical Equations - College Algebra

First, isolate the radical on one side of the equation. Start by adding to both sides of the equation.

Now, square both sides of the equation.

Expand the right side of the equation.

Collect all the terms to one side of the equation and simplify to create a quadratic equation equal to zero.

Quadratic equations can be written in the following generic form:

We need to find two numbers whose product equals a multiplied by c and whose sum equals b; therefore, the product of the factors must be -12 and their sum must equal -1. Notice the following:

Write the two quantities. When c is negative, one quantity needs to have a plus sign and one needs to have a minus sign. When b is negative, the larger number should be associated with the minus sign. Write the following quadratic factorization:

The solutions for this quadratic are:

One common mistake for students is to assume that these solutions are—in fact—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 solution actually works. As you will see, only one of the two solutions above actually works in this particular case.

We started with the following equation:

Now, substitute the solution into the equation.

We can see that -3 is not a solution. Now, substitute the solution into the equation.

The correct answer is 4.

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.

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.

Square both sides to eliminate the radical.

Divide by three on both sides to isolate the x.

The answer is:

Square both sides.

Add 2 on both sides.

Divide by 3 on both sides.

The answer is:

First, isolate the radical on one side of the equation. Start by adding to both sides of the equation.

Now, square both sides of the equation.

Expand the right side of the equation.

Collect all the terms to one side of the equation and simplify to create a quadratic equation equal to zero.

Quadratic equations can be written in the following generic form:

We need to find two numbers whose product equals a multiplied by c and whose sum equals b; therefore, the product of the factors must be -12 and their sum must equal -1. Notice the following:

Write the two quantities. When c is negative, one quantity needs to have a plus sign and one needs to have a minus sign. When b is negative, the larger number should be associated with the minus sign. Write the following quadratic factorization:

The solutions for this quadratic are:

One common mistake for students is to assume that these solutions are—in fact—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 solution actually works. As you will see, only one of the two solutions above actually works in this particular case.

We started with the following equation:

Now, substitute the solution into the equation.

We can see that -3 is not a solution. Now, substitute the solution into the equation.

The correct answer is 4.

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.

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.

Square both sides to eliminate the radical.

Divide by three on both sides to isolate the x.

The answer is:

Square both sides.

Add 2 on both sides.

Divide by 3 on both sides.

The answer is:

First, isolate the radical on one side of the equation. Start by adding to both sides of the equation.

Now, square both sides of the equation.

Expand the right side of the equation.

Collect all the terms to one side of the equation and simplify to create a quadratic equation equal to zero.

Quadratic equations can be written in the following generic form:

We need to find two numbers whose product equals a multiplied by c and whose sum equals b; therefore, the product of the factors must be -12 and their sum must equal -1. Notice the following:

Write the two quantities. When c is negative, one quantity needs to have a plus sign and one needs to have a minus sign. When b is negative, the larger number should be associated with the minus sign. Write the following quadratic factorization:

The solutions for this quadratic are:

One common mistake for students is to assume that these solutions are—in fact—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 solution actually works. As you will see, only one of the two solutions above actually works in this particular case.

We started with the following equation:

Now, substitute the solution into the equation.

We can see that -3 is not a solution. Now, substitute the solution into the equation.

The correct answer is 4.

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.

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.

Square both sides to eliminate the radical.

Divide by three on both sides to isolate the x.

The answer is:

Square both sides.

Add 2 on both sides.

Divide by 3 on both sides.

The answer is:

First, isolate the radical on one side of the equation. Start by adding to both sides of the equation.

Now, square both sides of the equation.

Expand the right side of the equation.

Collect all the terms to one side of the equation and simplify to create a quadratic equation equal to zero.

Quadratic equations can be written in the following generic form:

We need to find two numbers whose product equals a multiplied by c and whose sum equals b; therefore, the product of the factors must be -12 and their sum must equal -1. Notice the following:

Write the two quantities. When c is negative, one quantity needs to have a plus sign and one needs to have a minus sign. When b is negative, the larger number should be associated with the minus sign. Write the following quadratic factorization:

The solutions for this quadratic are:

One common mistake for students is to assume that these solutions are—in fact—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 solution actually works. As you will see, only one of the two solutions above actually works in this particular case.

We started with the following equation:

Now, substitute the solution into the equation.

We can see that -3 is not a solution. Now, substitute the solution into the equation.

The correct answer is 4.

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.

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.

Square both sides to eliminate the radical.

Divide by three on both sides to isolate the x.

The answer is:

Square both sides.

Add 2 on both sides.

Divide by 3 on both sides.

The answer is: