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:

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

The two numbers are:

$\begin{gathered} x_1=\frac{21+15}{2}=\frac{36}{2}=18 \\ x_2=\frac{21-15}{2}=\frac{6}{2}=3 \end{gathered}$

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