Question
How many real solutions does the equation have?
k^2 − 80 = –34
k^2 − 80 = –34
Answers
Answered by
GPT-4o mini
To solve the equation \( k^2 - 80 = -34 \), we first rearrange it to isolate \( k^2 \).
Starting with the original equation:
\[
k^2 - 80 = -34
\]
we add 80 to both sides:
\[
k^2 = -34 + 80
\]
Simplifying the right side:
\[
k^2 = 46
\]
To find \( k \), we take the square root of both sides:
\[
k = \pm \sqrt{46}
\]
Since the square root of a positive number has two solutions (one positive and one negative), we find that there are two real solutions for \( k \):
\[
k = \sqrt{46} \quad \text{and} \quad k = -\sqrt{46}
\]
Thus, the equation \( k^2 - 80 = -34 \) has \(\boxed{2}\) real solutions.
Starting with the original equation:
\[
k^2 - 80 = -34
\]
we add 80 to both sides:
\[
k^2 = -34 + 80
\]
Simplifying the right side:
\[
k^2 = 46
\]
To find \( k \), we take the square root of both sides:
\[
k = \pm \sqrt{46}
\]
Since the square root of a positive number has two solutions (one positive and one negative), we find that there are two real solutions for \( k \):
\[
k = \sqrt{46} \quad \text{and} \quad k = -\sqrt{46}
\]
Thus, the equation \( k^2 - 80 = -34 \) has \(\boxed{2}\) real solutions.