To solve the equation [tex]\( x^2 + 38 = 16 \)[/tex], let's go through the solution step-by-step:
1. Isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[
x^2 + 38 = 16
\][/tex]
First, subtract 38 from both sides of the equation:
[tex]\[
x^2 + 38 - 38 = 16 - 38
\][/tex]
Simplifying this gives:
[tex]\[
x^2 = -22
\][/tex]
2. Analyze the equation [tex]\( x^2 = -22 \)[/tex]:
In this step, observe that [tex]\( x^2 = -22 \)[/tex] implies that the square of [tex]\( x \)[/tex] equals a negative number.
In the field of real numbers, there is no real number that, when squared, results in a negative value. Therefore, there are no real solutions to this equation.
3. Conclusion for real numbers:
Because it's impossible for [tex]\( x^2 \)[/tex] to equal a negative number in the real number system, we conclude that this equation has no real solutions.
Thus, the final answer is:
[tex]\[
\text{This equation has no real solutions.}
\][/tex]
