Two numbers with given sum and product (II)

Text

Find two numbers whose sum is $11$ and whose product is $-20$.

Solution

By Vieta’s formulas, two numbers with sum $s$ and product $p$ are the roots of $t^2-s\,t+p=0.$ With $s=11$ and $p=-20$: $t^2-11t-20=0.$ Discriminant: $\Delta=(-11)^2-4\cdot1\cdot(-20)=121+80=201>0,$ so two distinct real solutions: $t=\frac{11\pm\sqrt{201}}{2}.$ Numerically, $\sqrt{201}\approx14.177$, hence $t_1=\frac{11+\sqrt{201}}{2}\approx12.589, \qquad t_2=\frac{11-\sqrt{201}}{2}\approx-1.589.$ Check: sum $t_1+t_2=11$; product $t_1 t_2=\dfrac{121-201}{4}=\dfrac{-80}{4}=-20$. ✓ $\boxed{x_1=\frac{11+\sqrt{201}}{2}\approx12.589,\qquad x_2=\frac{11-\sqrt{201}}{2}\approx-1.589}$