# length of rope (single variable equation)

The breaking strength of three-stand Manila rope is given by the formula $$L=5000d(d+1)$$ where L lb. is the load needed to snap a rope d inches in diameter. What rope will just snap under a load of 60000 lb.? --Sawyer, Mathematician's Delight

We know that $$60000=5000d^2+5000d$$ How can I solve the equation to calculate d?

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There is a general formula for solving quadratic equations, but this equation can be solved by factoring. Note that $$d^2 + d - 12 = (d + 4)(d - 3).$$ So, \begin{align}d^2 + d - 12 &= 0 \\ (d + 4)(d - 3) &= 0.\end{align} Notice that this is a product of two quantities: $d + 4$ and $d - 3$. For this product to be zero, then either of those must be zero, because if $xy = 0$, then either $x$ or $y$ (or both) must be zero. So, either $d + 4 = 0$ or $d - 3 = 0$. So, what's $d$?
This is a quadratc equation in $d$. Divide all thos big numbers away and bring all on one side to obtain the equation $$d^2+d-12=0.$$ Can you solve such a quadratic equation?