Question

(10 pts) The following is factored using the "smile method". x²-x-56. How do youknow this get determined? Cite at least TWO details that direct you to this method.Complete the first step of finding m and n: mn=m+n=m=n=

Answer 1

Solution:

The quadratic expression is given below as

$x^2-x-56$

Using the smiley method below , we will have it represents as

Let the two factors be represented below as

$m,n$

The product of the two factors will give us

$mn=-56-----(1)$

The sum of the two factors, will give

$\begin{gathered} m+n=-1----(2) \\ \end{gathered}$

from equation (2) make m the subject of the formula

$\begin{gathered} m+n=-1 \\ m=-1-n-----(3) \end{gathered}$

Substitute equation 3 in equation (1)

$\begin{gathered} mn=-56 \\ n(-1-n)=-56 \\ -n-n^2=-56 \\ -n^2-n=-56 \\ -n^2-n+56=0 \\ -n^2-8n+7n+56=0 \\ -n(n+8)+7(n+8)=0 \\ (7-n)(n+8)=0 \\ 7-n=0,n+8=0 \\ n=7,n=-8 \end{gathered}$

Substitute n=7 and n=-8 in equation (3)

$\begin{gathered} m=-1-n,when\text{ }n=7 \\ m=-1-7 \\ m=-8 \\ n=-1-n,whenn=-8 \\ m=-1-(-8) \\ m=-1+8=7 \end{gathered}$

Hence,

The final answers are

$\begin{gathered} \Rightarrow mn=-56 \\ \Rightarrow m+n=-1 \\ \Rightarrow m=7,n=-8 \\ \Rightarrow m=-8,n=7 \end{gathered}$