# Squaring both sides of an equation- extraneous solutions

• DS2C
In summary, when we square both sides of an equation, we can get extraneous solutions because squaring is not a reversible operation. If the operation applied to both sides is reversible, then there will not be any extraneous solutions. This can be seen in examples such as adding or subtracting the same quantity to both sides, or multiplying both sides by the same nonzero number. However, squaring both sides is not reversible, leading to extraneous solutions in some cases. This can also occur when raising both sides to an even power.
DS2C

## Homework Statement

Looking for an explanation as to why, when we square both sides of an equation, we can get extraneous solutions. That is, why can we square both sides of an equation, and sometimes the solutions we get are not true.

In my book, it gets a little wordy and doesn't make a lot of sense. It says that "If both sides of an equation are raised to the same power, all solutions of the original equation are among the solutions of the new equation. This does not say that raising both sides of an equation to a power yields an equivalent solution."
This makes no sense.

## Homework Equations

An example:
$$\sqrt {4 - x} = x - 2$$
$$\left(\sqrt{4 - x}\right)^2 = \left(x - 2\right)^2$$
$$x = 0 ~or~ x = 3$$

## The Attempt at a Solution

0 Results in a false statement where checked, and 3 results in a true statement.

Solution is {3}

But why does 0 result as an extraneous solution? We squared both sides of the equation, leaving it balanced.

DS2C said:
sometimes the solutions we get are not true.
Raising to an even power loses information. Both x and -x turn into x2. x=1 has one solution, x2=1 has two.

DS2C said:

## Homework Statement

Looking for an explanation as to why, when we square both sides of an equation, we can get extraneous solutions. That is, why can we square both sides of an equation, and sometimes the solutions we get are not true.

In my book, it gets a little wordy and doesn't make a lot of sense. It says that "If both sides of an equation are raised to the same power, all solutions of the original equation are among the solutions of the new equation. This does not say that raising both sides of an equation to a power yields an equivalent solution."
This makes no sense.

## Homework Equations

An example:$$\sqrt {4 - x} = x - 2$$ $$\left(\sqrt{4 - x}\right)^2 = \left(x - 2\right)^2$$ $$x = 0 ~or~ x = 3$$

## The Attempt at a Solution

0 Results in a false statement where checked, and 3 results in a true statement.

Solution is {3}

But why does 0 result as an extraneous solution? We squared both sides of the equation, leaving it balanced.
So, you might ask, "Why do we have this solution, ##\ x=0\,,\ ## which turns out to be extraneous?"

Plugging in ##\ x=0\ ## gives ##\ \sqrt{4}\ ## on the left hand side. There are two numbers which give 4 upon being squared, 2 and −2. If you pick the −2, then you match the right hand side. However, the radical symbol indicates that we only want the principal square root and that's what makes this solution be extraneous

Notice that squaring either of the following equations gives the same result.
## \sqrt {4 - x} = x - 2 ##

## -\sqrt {4 - x} = x - 2 ##​

##\ x=0\ ## is the solution to the second of these equations..

Let me see if I understand-
If we take ##2^2##, and ##\left(-2\right)^2##, we get 4 in both cases. However we don't know if 2 or -2 was used to get this 4 as it could have been either. Is this correct?

DS2C said:
Let me see if I understand-
If we take ##2^2##, and ##\left(-2\right)^2##, we get 4 in both cases. However we don't know if 2 or -2 was used to get this 4 as it could have been either. Is this correct?
Yes.

Ok thanks for the help guys. Very well explained.

If the operation you apply to both sides of the equation are reversible, then you won't get extaneous solutions. So adding the same quantity to both sides, subtracting the same quantify from both sides, multiplying both sides by the same nonzero number are examples of these kinds of operations. These are all reversible operations -- operations that are one-to-one, meaning that if the operations are represented as functions, these functions have inverses.

Squaring both sides is not an invertible operation, for the reasons already given. haruspex mentioned raising both sides to an even power as an operation that isn't invertible. If you raise both sides to an odd power, such as by cubing both sides, that's an operation that is invertible, provided we limit the discussion to the real numbers. For example, the equations ##x = 2## and ##x^3 = 8## are equivalent, for real numbers x. (If we also allow complex numbers as solutions, then the second equation has two more solutions than the first.)

Mark44 said:
If the operation you apply to both sides of the equation are reversible, then you won't get extaneous solutions. So adding the same quantity to both sides, subtracting the same quantify from both sides, multiplying both sides by the same nonzero number are examples of these kinds of operations. These are all reversible operations -- operations that are one-to-one, meaning that if the operations are represented as functions, these functions have inverses.

Squaring both sides is not an invertible operation, for the reasons already given. haruspex mentioned raising both sides to an even power as an operation that isn't invertible. If you raise both sides to an odd power, such as by cubing both sides, that's an operation that is invertible, provided we limit the discussion to the real numbers. For example, the equations ##x = 2## and ##x^3 = 8## are equivalent, for real numbers x. (If we also allow complex numbers as solutions, then the second equation has two more solutions than the first.)
Move to my university and teach my math classes.

DS2C said:
Move to my university and teach my math classes.
I'll be down your way next week, but just for a visit. I'm already teaching part-time at a CC in Washington state.

## 1. What does it mean to square both sides of an equation?

Squaring both sides of an equation refers to the process of raising both sides of an equation to the power of 2. This is done in order to eliminate radicals or to solve for a variable.

## 2. Why is it important to check for extraneous solutions when squaring both sides of an equation?

When squaring both sides of an equation, it is possible to introduce extraneous solutions, which are solutions that do not actually satisfy the original equation. Checking for extraneous solutions is important to ensure that the solutions obtained are valid for the original equation.

## 3. How do you check for extraneous solutions when squaring both sides of an equation?

To check for extraneous solutions, you must substitute the solutions obtained back into the original equation and confirm that they satisfy the equation. If the substituted value does not satisfy the original equation, it is an extraneous solution.

## 4. What are some common mistakes made when squaring both sides of an equation?

Some common mistakes include forgetting to square both sides, making errors when simplifying the squared terms, and not checking for extraneous solutions. It is important to double-check the steps and solutions when squaring both sides of an equation to avoid these mistakes.

## 5. Are there any exceptions when squaring both sides of an equation?

Yes, there are some exceptions when squaring both sides of an equation. If the original equation contains variables in the denominator, the solutions obtained after squaring both sides may not satisfy the original equation. In this case, the extraneous solutions cannot be eliminated and must be included in the final solution set.

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