Two numbers with given sum and product

Text

Find two numbers whose sum is $5$ and whose product is $-4$.

Solution

By Vieta’s formulas, two numbers with sum $s$ and product $p$ are the roots of the equation $t^2-s\,t+p=0.$ With $s=5$ and $p=-4$: $t^2-5t-4=0.$ Discriminant: $\Delta=(-5)^2-4\cdot1\cdot(-4)=25+16=41>0,$ so there are two distinct real solutions: $t=\frac{5\pm\sqrt{41}}{2}.$ Numerically, $\sqrt{41}\approx6.403$, hence $t_1=\frac{5+\sqrt{41}}{2}\approx5.702, \qquad t_2=\frac{5-\sqrt{41}}{2}\approx-0.702.$ Check: sum $t_1+t_2=5$; product $t_1 t_2=\dfrac{25-41}{4}=\dfrac{-16}{4}=-4$. ✓ $\boxed{x_1=\frac{5+\sqrt{41}}{2}\approx5.702,\qquad x_2=\frac{5-\sqrt{41}}{2}\approx-0.702}$