Question

Find two numbers whose sum is 27 and product is 182.

Answer 1

Hint: At first take a number as x so the other number is $27-x$ . Then write it as –
$x\left( 27-x \right)=182$ and solve the quadratic equation to solve for x.

Complete step-by-step answer:
In the question we are told that there are two numbers whose sum is 27 and product is 182 and we have to find those two numbers.
As we know that the sum of two numbers is 27, so if one of them is x then the other will be $27-x$ . So, the numbers are x and $27-x$ . We are given that the product of two numbers is 182 so, we can write as –
$x\left( 27-x \right)=182$ .
Now on expanding we can write it as, $27x-{{x}^{2}}=182$ .
On rearranging the equation we get, ${{x}^{2}}-27x+182=0$ .
Now, we will further write equations as –
 ${{x}^{2}}-13x-14x+182=0$ . So, we get on factorising,
 $x\left( x-13 \right)-14\left( x-13 \right)=0$ .
or, $\left( x-14 \right)\left( x-13 \right)=0$ .
So, the value of x is either 14 or 13. If $x=14$ then other number will be $\left( 27-x \right)$ or \[27-14=13\] and if \[x=13\] then other number will be $\left( 27-x \right)$ or \[27-13=14\] .
So, the number which satisfies the given condition in the question is 13 and 14.

Note:- We can also do it in other way by taking number as x , y and write it as \[x\text{ }+\text{ }y\text{ }=\text{ }27\] and \[x\text{ }y=182\] according to the given condition in the question. So, we will find the value of \[x\text{ }\text{ }y\] from value of $\left( x+y \right)$ and x y using formula ${{\left( x+y \right)}^{2}}={{\left( x-y \right)}^{2}}+4xy$ and therefore solve equation to get x and y.