Question

write two numbers that multiply to the value on top and add to the value on bottom.
answer
attempt 1 out of 2
27
-90

Explanation:

Step1: Let the two numbers be \(x\) and \(y\).

We know that \(xy=- 90\) and \(x + y=27\). From \(x + y=27\), we have \(y = 27 - x\). Substitute \(y = 27 - x\) into \(xy=-90\), getting \(x(27 - x)=-90\).

Step2: Expand the equation.

Expand \(x(27 - x)=-90\) to \(27x-x^{2}=-90\), then rewrite it as \(x^{2}-27x - 90=0\).

Step3: Use the quadratic - formula.

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 1\), \(b=-27\), \(c=-90\)), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). First, calculate the discriminant \(\Delta=b^{2}-4ac=(-27)^{2}-4\times1\times(-90)=729 + 360=1089\). Then \(x=\frac{27\pm\sqrt{1089}}{2}=\frac{27\pm33}{2}\).

Step4: Find the two solutions.

When we take the plus - sign: \(x=\frac{27 + 33}{2}=\frac{60}{2}=30\). When we take the minus - sign: \(x=\frac{27-33}{2}=\frac{-6}{2}=-3\). If \(x = 30\), then \(y=27 - 30=-3\); if \(x=-3\), then \(y=27-(-3)=30\).

Answer:

30, - 3