Question

For the following two numbers, find two factors of the first number such that their product is the first number and their sum is the second number,40, 14

Answer 1

First we need to factorate the number 40:

`40=2\cdot2\cdot2\cdot5`

The possible numbers we can create using these factors are 2, 4, 5, 8, 10 and 20.

So If the product of the two factors (let's call them 'a' and 'b') is 40 and the sum is 14, we have:

`\begin{gathered} a\cdot b=40 \\ a+b=14 \\ \\ \text{From the second equation:} \\ b=14-a \\ \\ \text{Using this value of b in the first equation:} \\ a(14-a)=40 \\ 14a-a^2=40 \\ a^2-14a+40=0 \end{gathered}`

Using the quadratic formula to solve this equation, we have:

`\begin{gathered} a_1=\frac{-b+\sqrt[]{b^2-4ac}}{2a}=\frac{14+\sqrt[]{196-160}}{2}=(14+6)/(2)=10 \\ a_2=\frac{-b-\sqrt[]{b^2-4ac}}{2a}=(14-6)/(2)=4 \\ \\ a=10\to b=14-10=4 \\ a=4\to b=14-4=10 \end{gathered}`

So the factors which product is 40 and the sum is 14 are 4 and 10.