What are the solutions to the equation x^2 = 169?
The Correct Answer and Explanation is:
To solve the equation ( x^2 = 169 ), we need to isolate ( x ) by taking the square root of both sides.
Step-by-Step Solution:
- Start with the equation:
[
x^2 = 169
]
The goal is to find the values of ( x ) that satisfy this equation. - Take the square root of both sides:
To solve for ( x ), we apply the square root operation to both sides of the equation. Remember that when you take the square root of both sides, you must consider both the positive and negative roots because squaring either a positive or negative number results in a positive value. Mathematically, this looks like:
[
\sqrt{x^2} = \pm \sqrt{169}
] - Simplify the square root:
The square root of 169 is 13 because:
[
\sqrt{169} = 13
]
Therefore, the equation becomes:
[
x = \pm 13
]
This means that ( x ) can be either 13 or -13.
Conclusion:
The solutions to the equation ( x^2 = 169 ) are:
[
x = 13 \quad \text{or} \quad x = -13
]
Explanation:
The square of any number ( x ) (whether positive or negative) results in the same positive value. For example:
- ( 13^2 = 169 )
- ( (-13)^2 = 169 )
This is why both 13 and -13 are solutions to the equation ( x^2 = 169 ). When solving quadratic equations like this, it’s important to remember that squaring a number eliminates the sign, so both the positive and negative roots must be considered.
Thus, for any equation of the form ( x^2 = a ), where ( a ) is a positive number, there will typically be two solutions: one positive and one negative.