1. **State the problem:** We are given the bending moment formula for a beam: $$ M = \frac{w \ell}{2} x - \frac{w}{2} x^2 $$ We need to find the value of $x$ when $\ell = 13$, $w = 16$, and $M = 288$. 2. **Substitute the known values into the formula:** $$ 288 = \frac{16 \times 13}{2} x - \frac{16}{2} x^2 $$ 3. **Simplify the coefficients:** $$ 288 = \frac{208}{2} x - 8 x^2 $$ $$ 288 = 104 x - 8 x^2 $$ 4. **Rewrite the equation in standard quadratic form:** $$ 0 = 104 x - 8 x^2 - 288 $$ $$ 0 = -8 x^2 + 104 x - 288 $$ 5. **Multiply both sides by $-1$ to make the leading coefficient positive:** $$ 0 = 8 x^2 - 104 x + 288 $$ 6. **Divide the entire equation by 8 to simplify:** $$ 0 = \cancel{8} x^2 - \cancel{104} x + \cancel{288} \div 8 $$ $$ 0 = x^2 - 13 x + 36 $$ 7. **Solve the quadratic equation $x^2 - 13 x + 36 = 0$ using the quadratic formula:** The quadratic formula is: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=-13$, and $c=36$. 8. **Calculate the discriminant:** $$ \Delta = (-13)^2 - 4 \times 1 \times 36 = 169 - 144 = 25 $$ 9. **Find the roots:** $$ x = \frac{-(-13) \pm \sqrt{25}}{2 \times 1} = \frac{13 \pm 5}{2} $$ 10. **Calculate each solution:** - $$ x = \frac{13 + 5}{2} = \frac{18}{2} = 9 $$ - $$ x = \frac{13 - 5}{2} = \frac{8}{2} = 4 $$ **Final answer:** The value of $x$ can be either $4$ or $9$ meters.