Find all real square roots of 25
The Correct Answer and Explanation is:
The square root of a number ( x ) is a value ( y ) that satisfies the equation ( y^2 = x ). Specifically, we are looking for the real square roots of 25. In mathematical terms, this means we are seeking all values of ( y ) such that:
[
y^2 = 25
]
To solve for ( y ), we take the square root of both sides of the equation:
[
y = \pm \sqrt{25}
]
This expression indicates that there are two possible values of ( y ) since squaring either a positive or a negative number results in the same positive value. In this case, we know:
[
\sqrt{25} = 5
]
Thus, the two possible solutions are:
[
y = 5 \quad \text{or} \quad y = -5
]
Therefore, the real square roots of 25 are ( \pm 5 ).
Explanation
- The Concept of a Square Root:
The square root of a number is defined as the number that, when multiplied by itself, gives the original number. For example, ( \sqrt{25} ) is the number that, when multiplied by itself, equals 25. - Why Two Solutions?:
For positive real numbers, the square root operation has two possible solutions: a positive root and a negative root. This is because both ( 5 \times 5 = 25 ) and ( (-5) \times (-5) = 25 ). Therefore, both ( 5 ) and ( -5 ) are valid square roots of 25. - Positive vs. Negative Roots:
When we take the square root of a number in basic arithmetic, it typically refers to the principal (non-negative) root. However, the general solution for square roots includes both the positive and negative values that satisfy the equation. - Conclusion:
The real square roots of 25 are ( 5 ) and ( -5 ). These are the two real solutions to the equation ( y^2 = 25 ), both of which are correct answers.