Question

write two numbers that multiply to the value on top and add to the value on bottom. 1 + × -90

Explanation:

Step1: Set up equations

Let the two numbers be $x$ and $y$. We have $xy=- 90$ and $x + y=1$. From $x + y=1$, we get $y = 1 - x$. Substitute $y = 1 - x$ into $xy=-90$, we get $x(1 - x)=-90$.

Step2: Rearrange the equation

Expand $x(1 - x)=-90$ to $x - x^{2}=-90$, then rearrange it to the quadratic - form $x^{2}-x - 90=0$.

Step3: Factor the quadratic equation

We need to find two numbers that multiply to $-90$ and add to $-1$. The numbers are $-10$ and $9$ since $(-10)\times9=-90$ and $-10 + 9=-1$. So, $x^{2}-x - 90=(x - 10)(x+9)=0$.

Step4: Solve for $x$

Set each factor equal to zero: $x - 10 = 0$ gives $x = 10$; $x + 9 = 0$ gives $x=-9$. When $x = 10$, $y=1 - 10=-9$; when $x=-9$, $y=1-(-9)=10$.

Answer:

10 and - 9