Question

what two numbers add to 5 and multiply to -3

Answer 1

Answer:

$x = \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$   and   $y = \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$

$x = \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$   and   $y = \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$

Step-by-step explanation:

Let the numbers be x and y.

We now have a system of equations.

x + y = 5

xy = -3

Solve the second equation for x.

x = -3/y

Now substitute x in the first equation with -3/y.

-3/y + y = 5

Multiply both sides by y.

-3 + y^2 = 5y

y^2 - 5y - 3 = 0

Use the quadratic formula to solve for y.

$y = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$y = \dfrac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-3)}}{2(1)}$

$y = \dfrac{5 \pm \sqrt{25 + 12}}{2}$

$y = \dfrac{5 \pm \sqrt{37}}{2}$

$y = \dfrac{5 + \sqrt{37}}{2}$   or   $y = \dfrac{5 - \sqrt{37}}{2}$

$y = \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$   or   $y = \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$

We get 2 solutions for y. Now for each solution for y, we need to find a corresponding solution for x.

Solve the first equation for x.

x + y = 5

x = 5 - y

Substitute each y value to find the corresponding x value.

$x = 5 - (\dfrac{5}{2} + \dfrac{\sqrt{37}}{2})$

$x = 5 - \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$

$x = \dfrac{10}{2} - \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$

$x = \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$

This give one solution as:

$x = \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$   and   $y = \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$

Now we substitute the other y value to find the other x value.

$x = 5 - (\dfrac{5}{2} - \dfrac{\sqrt{37}}{2})$

$x = 5 - \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$

$x = \dfrac{10}{2} - \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$

$x = \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$

This give the second solution as:

$x = \dfrac{5}{2} + \dfrac{\sqrt{37}}{2}$   and   $y = \dfrac{5}{2} - \dfrac{\sqrt{37}}{2}$