Take the square root of a variable that is squared?

[yo4561]

Let's say I have x^2, and I take the square root of it. Why is the solution the absolute value of x?

[TiffanyB]

Let's say I have x^2, and I take the square root of it. Why is the solution the absolute value of x?

Hello yo4561,

Let's look at an example to demonstrate this.

Example: x^2 = 4

In this case, what is x? Well, the most obvious answer is that x could be 2.

Could it be anything else? Yes! x could also be -2, because -2*-2 = 4.

Using this example, x could be 2 or -2.

However, when you are taking the square room of a number, as in "what is the square root of 25?" that answer is always the positive result. In this case, you can think of sqrt (4) = 2.

To recap:

If x^2 = 4, what is x? Answer could be 2 or -2.
What is sqrt (x) if x is 4? Answer is 2 only (not -2).

[JbhB682]

Let's say I have x^2, and I take the square root of it. Why is the solution the absolute value of x?

Hello yo4561,

Let's look at an example to demonstrate this.

Example: x^2 = 4

In this case, what is x? Well, the most obvious answer is that x could be 2.

Could it be anything else? Yes! x could also be -2, because -2*-2 = 4.

Using this example, x could be 2 or -2.

However, when you are taking the square room of a number, as in "what is the square root of 25?" that answer is always the positive result. In this case, you can think of sqrt (4) = 2.

To recap:

If x^2 = 4, what is x? Answer could be 2 or -2.
What is sqrt (x) if x is 4? Answer is 2 only (not -2).

Hi Tiffany - you mention per the red that in both scenario's
scenario 1) sqrt {(-2)^2}
scenario 2) sqrt {(+2)^2}

The value of x is +2 only (not -2) (in scenario #1 and scenario # 2 )

By that same trend / logic
scenario 3) sqrt {(-x)^2}
scenario 4) sqrt {(x)^2}

the value of x should be +x ONLY (not -x) (in scenario 3 and scenario 4)

Why do you think the trend is NOT PLAYING out when variables are involved but not in the case of actual numbers.

[TiffanyB]

Hello JbhB682,

The trend that I described applies both to real numbers and to variables. When the GMAT gives you a square root symbol, only use the positive root. This is true for any even roots (e.g., square root, 4th root, etc).

When the GMAT gives a variable that is squared, such as x^2, then consider that x could be positive or negative.

Note: The information if you have an odd root symbol, such as a a cube root, 5th root, etc. and a negative base. For the example, the cube root of -27 would be -3.