1. **State the problem:** We are given the sag formula $$l = \frac{12}{d} + d$$ where $l$ is the sag in metres and $d$ is the distance between the supports in metres. 2. We need to find the distance $d$ when the sag $l$ is equal to $20 + x$ metres. 3. Substitute $l = 20 + x$ into the formula: $$20 + x = \frac{12}{d} + d$$ 4. Multiply both sides by $d$ to clear the denominator: $$d(20 + x) = 12 + d^2$$ 5. Rearrange the equation to standard quadratic form: $$d^2 - d(20 + x) + 12 = 0$$ 6. This is a quadratic equation in $d$: $$d^2 - (20 + x)d + 12 = 0$$ 7. Use the quadratic formula to solve for $d$: $$d = \frac{(20 + x) \pm \sqrt{(20 + x)^2 - 4 \times 1 \times 12}}{2}$$ 8. Simplify the discriminant: $$\sqrt{(20 + x)^2 - 48}$$ 9. Therefore, the two possible distances between the supports are: $$d = \frac{(20 + x) \pm \sqrt{(20 + x)^2 - 48}}{2}$$ 10. Choose the positive value(s) of $d$ that make physical sense (distance must be positive).