Let's solve the equation [tex]\((x-3)^2 = -4\)[/tex] using the square root property of equality.
1. Understanding the Equation:
The given equation is [tex]\((x-3)^2 = -4\)[/tex]. Here, [tex]\((x-3)\)[/tex] is squared, and we are expected to equal that to [tex]\(-4\)[/tex].
2. Square Root Property:
The square root property of equality states that if [tex]\(a^2 = b\)[/tex], then [tex]\(a = \pm \sqrt{b}\)[/tex].
3. Applying the Square Root Property:
We try to take the square root of both sides of the equation:
[tex]\[
(x-3)^2 = -4
\][/tex]
To solve for [tex]\(x-3\)[/tex], we take the square root of both sides:
[tex]\[
x-3 = \sqrt{-4}
\][/tex]
4. Square Root of a Negative Number:
Here, however, the square root property encounters an issue. The square root of [tex]\(-4\)[/tex] is not a real number because the square root of any negative number is not defined within the set of real numbers.
The square of any real number is always non-negative, meaning that it cannot be negative. As a result, there are no real numbers [tex]\(x\)[/tex] such that [tex]\((x-3)^2 = -4\)[/tex].
5. Conclusion:
Therefore, examining the equation [tex]\((x-3)^2 = -4\)[/tex], we conclude that there are no real solutions because a squared term can never equal a negative number.
In conclusion:
[tex]\[
\boxed{\text{There are no real solutions for the equation } (x-3)^2 = -4.}
\][/tex]
