Question

What two numbers would equal 8 and -3 when adding and multiplying them?

Answer 1

Let us assume first number is = x and second number is =y.

Adding those numbers we get 8. So we can setup first equation as

x+y=8. ..........................equation (1)

Multiplying those numbers we get -3. So we can setup second equation as

x*y=-3. ..........................equation(2).

we need to solve first equation for y

x+y=8 subtracting x from both sides, we get

x-x+y=8-x

y=8-x.

Substituting x=8-x in second equation, we get

x*(8-x)=-3.

Distributing x over (8-x), we get

8x -x^2 =-3

Adding 3 on both sides, we get

8x -x^2+3 =-3+3

-x^2 + 8x +3 = 0

We can solve this quadratic equation for x now.

We have a=-1, b=8 and c=3.

Plugging values of a, b and c in quadratic formula, we get

`x=(-b\pm √(b^2-4ac))/(2a)`

`x=(-8\pm √(8^2-4\left(-1\right)3))/(2\left(-1\right))`

`[=(-8+√(76))/(-2)`and

`x=(-8-√(76))/(-2)`

`\mathrm{Factor}\:8-2√(19):\quad 2\left(4-√(19)\right)`

We get

`x=(2\left(4-√(19)\right))/(2)` and

`x=(2\left(4+√(19)\right))/(2)`

On simplfying above two values, we get

`x=-√(19)+4,\:x=4+√(19)`

So, the required numbers are

`-√(19)+4 \ and \ 4+√(19)`.