Question

If a number is added to its square, the result is $ 42 $ . How do you find the number?

Answer 1

Hint: We are given a statement and we have to find the value of the number according to the statement. First we will assume the number as any variable for example let the number be x. Then we will form the equation according to whatever it is said after the equation is formed. It will be a quadratic equation then we will solve it using splitting the middle term method. We will find the value of x.

Complete step by step answer:
Step1:
Given that a number is added to its square the result is $ 42 $ . Let’s assume the number be x and form the equation as
 $ {x^2} + x = 42 $
The equation so formed is a quadratic equation. Then we will rearrange the quadratic equation. On rearrangement we will get:
 $ \Rightarrow {x^2} + x - 42 = 0 $

Step2:
Now we will use the splitting the middle term method to solve this equation. First we will factor $ 42 $ then we get the factor $ 6 \text{and} 7 $ . So we will split the middle term as the difference of $ 7x ; 6x $ .
 $ \Rightarrow {x^2} + 7x - 6x - 42 = 0 $
Factor out common terms in first two terms we get:
 $ \Rightarrow x\left( {x + 7} \right) - 6x - 42 = 0 $
Factor out common terms in next two terms we will get:
 $ \Rightarrow x\left( {x + 7} \right) - 6\left( {x + 7} \right) = 0 $
Now doing the factorization of the terms we will get:
 $ \Rightarrow \left( {x - 6} \right)\left( {x + 7} \right) = 0 $

Step3:
Equating each term equal to zero we will get:
 $ \Rightarrow x - 6 = 0 $
 $ x = 6 $ ;
 $ \Rightarrow x + 7 = 0 $
 $ x = - 7 $

Note: First assume the variable and write the equation as the statement says. The formation of quadratic equations is quite easy because it involves only one variable. Sometimes equation form is big and complicated then it can be easily solved by quadratic formula and we get the values of the variable by solving it. We get the two values of the variable sometimes one is negative and one is positive so we choose it according to the requirement of the question.