Question

The product of two numbers is 225, but their sum is -30. What are these two numbers?

Answer 1

The prompt states two equations in the first sentence. Make words in sentence a math equation.

$xy=225\\x+y=-30$

You now have two equations and two unknowns (x and y). Solve for x (or y, it doesn't matter).

$x+y=-30\\y=-30-x$

Substitute this equation into the second equation and solve for x.

$xy=225\\x(-30-x)=225$

After rearranging the terms you realize that the equation is a quadratic so use the quadratic formula to find the roots (also known as "zeros")

$-x^{2}-30x-225=0\\quadratic formula \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$^{2}=225\\[/tex]

where

$a=-1\\b=-30\\c=-225$

so

$\frac{30\pm\sqrt{-30^{2}-4(-1)(-225)} } {2(-1)}=-15$

therefore

$x=-15$

Now that you found x substitute into one of the original two equations.

$x+y=-30\\-15+y=-30\\y=-30+15\\y=-15$

You've found the two numbers.