If we know the sum and product of two numbers, what are they?

If we know the sum and product of two numbers, what are they?

Question

The sum and product of two numbers, $x$ and $y$, are $10$ and $12$ respectively. Form appropriate equations, plot [on graph paper] the corresponding graphs and hence obtain approximate values for $x$ and $y$.

Interactive diagram

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The red rectangle has a horizontal side of length $x$.

The vertical side has length $y$, and the perimeter is fixed at $20$. The area of the red rectangle is shown.

Solution

The information given in the question tells us that

$$\begin{align*} x + y &= 10, \\ xy &= 12. \end{align*}$$

By rearranging these equations to make

$y$

the subject, we get

$$\begin{align*} y &= 10 - x, \\ y &= \frac{12}{x}. \end{align*}$$

The following graph shows both the line (corresponding to the first equation) and the hyperbola (corresponding to the second equation).

The graph shows that there is an approximate solution with $x \approx 1.4$ and $y \approx 8.6$ and another one with $x \approx 8.6$ and $y \approx 1.4$. These give essentially the same solution, as we are looking for two numbers with a given sum and given product.

The equations can be solved exactly, and without much difficulty. By substituting the equation

$y = 10 - x$

into the equation

$xy = 12$

, we have

$$x(10 - x) = 12$$

so that

$$x^2 - 10x + 12 = 0.$$

By the

quadratic formula

, we have that

$$\begin{align*} x &= \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} \\ &= \frac{10 \pm \sqrt{100 - 48}}{2} \\ &= \frac{10 \pm \sqrt{52}}{2}. \end{align*}$$

The smaller of these values is $1.394448\!\ldots$, which is close to our visual estimate.