200/(15+x) + 200/(15-x) = 200

Let's solve the equation step by step: Given: \[ \frac{200}{15 + x} + \frac{200}{15 - x} = 200 \] **Step 1: Find a common denominator** The denominators are \(15 + x\) and \(15 - x\). The common denominator is \((15 + x)(15 - x)\). **Step 2: Write the equation with a common denominator** \[ \frac{200(15 - x)}{(15 + x)(15 - x)} + \frac{200(15 + x)}{(15 + x)(15 - x)} = 200 \] Combine the numerators: \[ \frac{200(15 - x) + 200(15 + x)}{(15 + x)(15 - x)} = 200 \] **Step 3: Simplify numerator** \[ 200(15 - x) + 200(15 + x) = 200 \times 15 - 200x + 200 \times 15 + 200x \] \[ = (200 \times 15 + 200 \times 15) + (-200x + 200x) = 2 \times 200 \times 15 + 0 = 2 \times 200 \times 15 \] Calculate: \[ 2 \times 200 \times 15 = 2 \times 200 \times 15 = 2 \times 3000 = 6000 \] **Step 4: Rewrite the equation** \[ \frac{6000}{(15 + x)(15 - x)} = 200 \] Recall that: \[ (15 + x)(15 - x) = 15^2 - x^2 = 225 - x^2 \] So, the equation becomes: \[ \frac{6000}{225 - x^2} = 200 \] **Step 5: Cross-multiplied** \[ 6000 = 200 (225 - x^2) \] Divide both sides by 200: \[ \frac{6000}{200} = 225 - x^2 \] \[ 30 = 225 - x^2 \] **Step 6: Solve for \(x^2\)** \[ x^2 = 225 - 30 = 195 \] **Step 7: Find \(x\)** \[ x = \pm \sqrt{195} \] Simplify \(\sqrt{195}\): \[ 195 = 39 \times 5 \] So: \[ x = \pm \sqrt{39 \times 5} = \pm \sqrt{39} \times \sqrt{5} \] If needed, approximate: \[ \sqrt{39} \approx 6.245, \quad \sqrt{5} \approx 2.236 \] \[ x \approx \pm 6.245 \times 2.236 \approx \pm 13.96 \] --- ### **Final answer:** \[ \boxed{ x = \pm \sqrt{195} \approx \pm 13.96 } \]