Question

Divide \[12\] into two parts such that their product is \[32\] .

Answer 1

Hint: Here, in the given question, we are asked to divide \[12\] into two such parts that their product comes out to be \[32\] . To solve this question, we will take two parts of \[12\] as two unknown variables, \[x\] and \[y\] . This will help us in forming two linear equations with two variables as we already know the sum and the product of both the unknown parts. As we also know that, two equations are needed to solve for two unknown variables, we will continue to solve and reach the desired answer.

Complete step-by-step answer:
We have to divide \[12\] into two parts such that the product is \[32\] .
Let us assume that two parts of \[12\] such that their product is \[32\] be \[x\] and \[y\] .
Then, \[x + y = 12\] \[ \ldots \left( 1 \right)\]
And, \[xy = 32\] \[ \ldots \left( 2 \right)\]
Now, use the substitution method to solve the system of linear equations.
 \[ \Rightarrow x = \dfrac{{32}}{y}\]
Put the value of \[x = \dfrac{{32}}{y}\] in equation \[\] ,
 \[\dfrac{{32}}{y} + y = 12\]
Multiplying \[y\] both sides, we get,
 \[
  32 + {y^2} = 12y \\
   \Rightarrow {y^2} - 12y + 32 = 0 \\
 \]
Solve for \[y\] using the middle term splitting method of factorization.
 \[
  {y^2} - 4y - 8y - 32 = 0 \\
   \Rightarrow y\left( {y - 4} \right) - 8\left( {y - 4} \right) = 0 \\
 \]
Taking \[\left( {y - 4} \right)\] common, we get,
 \[\left( {y - 4} \right)\left( {y - 8} \right) = 0\]
Now, the product of two numbers can be zero, only and only if at-least one of the numbers is zero.
If \[\left( {y - 4} \right) = 0\] ,
 \[ \Rightarrow y = 4\]
And, if \[\left( {y - 8} \right) = 0\]
 \[ \Rightarrow y = 8\]
Solve for \[x\] using the value of \[y\] using equation \[\left( 1 \right)\]
When \[y = 4\] ,
 \[x = 8\]
When \[y = 8\] ,
 \[x = 4\]
Hence, we can conclude that the two required numbers are \[4,8\] .

Note: Alternatively, this question can also be solved using one variable only. In that case, if one of the numbers is \[x\] , then the other number will be \[\left( {12 - x} \right)\] . Now, from here we can multiply them to equate with \[32\] and directly form a quadratic equation and then solve in a similar way using the factorization method.