Question

How do you find the value of C that makes \[{x^2} - 24x + C\] into a perfect square?

Answer 1

Hint: We are given with an equation or we can say an identity that is to be treated as a perfect square and we need to find the value of C such that it makes the above identity a perfect square. For comparison we will use the identity \[{\left( {x - y} \right)^2} = {x^2} - 2xy + {y^2}\].

Complete step-by-step answer:
Given that \[{x^2} - 24x + C\] is a given equation
But we can write this as \[{x^2} - 2 \times x \times 12 + C\]
Now this is same as \[{x^2} - 2xy + {y^2}\]
Thus we have here the value of y as 12. Thus value of C is \[{y^2}\]
Thus \[{y^2} = {12^2} = 144\]
Thus the value of C is 144. Now putting this in the equation given we can write
\[{x^2} - 2 \times x \times 12 + 144\]
This is a perfect square of \[{\left( {x - 12} \right)^2}\]. Thus the value of C that makes the equation perfect square is 144.
So, the correct answer is “144”.

Note: Note that we have to find the value of C by considering the equation is a perfect square. Thus we will split the terms as that of perfect square such that the third term should be a perfect square definitely. We just have to find the value of C and not the roots of the equation given above.