Solve the absolute value equation. $$ \left|x^{2}-36\right|=35 $$

Question

UpStudy AI · Accepted Answer

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:**

$$
x = \pm 1 \quad \text{and} \quad x = \pm \sqrt{71}
$$
The solutions are $$ x = 1, -1, \sqrt{71}, -\sqrt{71} $$.

Solve the absolute value equation. \( \left|x^{2}-36\right|=35 \)