X to the power of 2= 25 what does x equal
The Correct Answer and Explanation is:
The equation provided is ( x^2 = 25 ).
Solving for ( x ):
To solve for ( x ), we need to isolate ( x ) on one side of the equation. To do this, we can take the square root of both sides of the equation.
[
x^2 = 25
]
Take the square root of both sides:
[
\sqrt{x^2} = \sqrt{25}
]
Since the square root of ( x^2 ) is ( |x| ) (the absolute value of ( x )):
[
|x| = 5
]
This means ( x ) can be either positive or negative because both ( 5^2 ) and ( (-5)^2 ) will give 25.
Therefore, ( x ) can be:
[
x = 5 \quad \text{or} \quad x = -5
]
Explanation:
The equation ( x^2 = 25 ) is a quadratic equation. Quadratic equations generally have two solutions because squaring a positive number and squaring its negative counterpart results in the same value. This is an important concept when solving for variables in equations involving squares.
When you square a number, you’re essentially multiplying the number by itself, and this process “loses” the information about whether the original number was positive or negative. For instance, if ( x = 5 ), then ( x^2 = 25 ), but if ( x = -5 ), ( (-5)^2 = 25 ) as well. This demonstrates why both positive and negative values for ( x ) are solutions.
In real-world contexts, this idea applies to situations where both directions (positive and negative) can yield the same result. For example, if you’re looking for the value of a number whose square is 25, you would need to account for both positive 5 and negative 5.
Thus, the two solutions to ( x^2 = 25 ) are:
[
x = 5 \quad \text{or} \quad x = -5
]