Question
Solve the absolute value equation. \( \left|x^{2}-36\right|=35 \)
Ask by Conner Ray. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = 1, -1, \sqrt{71}, -\sqrt{71} \).
Solution
To solve the absolute value equation:
\[
|x^2 - 36| = 35
\]
We need to consider the two scenarios that make the absolute value equal to 35:
1. **Positive Case:**
\[
x^2 - 36 = 35
\]
\[
x^2 = 35 + 36
\]
\[
x^2 = 71
\]
\[
x = \pm \sqrt{71}
\]
2. **Negative Case:**
\[
x^2 - 36 = -35
\]
\[
x^2 = -35 + 36
\]
\[
x^2 = 1
\]
\[
x = \pm 1
\]
**Final Solutions:**
\[
x = \pm 1 \quad \text{and} \quad x = \pm \sqrt{71}
\]
**So, the solutions are:**
\[
x = \pm 1 \quad \text{and} \quad x = \pm \sqrt{71}
\]
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Simplify this solution Bonus Knowledge
To solve the absolute value equation \( \left|x^{2}-36\right|=35 \), we can break it down into two separate cases based on the definition of absolute value. **Case 1:** \( x^{2} - 36 = 35 \) Adding 36 to both sides: \( x^{2} = 71 \) Taking the square root of both sides: \( x = \sqrt{71} \) or \( x = -\sqrt{71} \) **Case 2:** \( x^{2} - 36 = -35 \) Adding 36 to both sides: \( x^{2} = 1 \) Taking the square root of both sides: \( x = 1 \) or \( x = -1 \) So, the solutions to the equation \( \left|x^{2}-36\right|=35 \) are: \( x = \sqrt{71}, -\sqrt{71}, 1, -1 \)
