Question

Write two numbers that multiply to 54 and add to -21

Answer 1

This is the solution to a 2 degree equation:

`x^2-21x+54=0`

The two numbers that multiply to 54 and add to -21 are the roots of this equation.

We can find the solution of an equation with this form:

`ax^2+bx+c=0`

With this formula:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

For this problem, a = 1, b = -21 and c = 54:

`x=\frac{21\pm\sqrt[]{(-21)^2-4\cdot1\cdot54}}{2\cdot1}=\frac{21\pm\sqrt[]{441-216}}{2}=\frac{21\pm\sqrt[]{225}}{2}=(21\pm15)/(2)`

The two numbers are:

`\begin{gathered} x_1=(21+15)/(2)=(36)/(2)=18 \\ x_2=(21-15)/(2)=(6)/(2)=3 \end{gathered}`

The numbers are -18 and -3 (they must be negative so they add up to a negative number)