Question

Find the value of ‘c’ that makes the trinomial $ \left( {{x^2} + 9x + c} \right) $ a perfect square?

Answer 1

Hint: In the given question, we are required to find the value of a variable c such that the trinomial given to us in the question as $ \left( {{x^2} + 9x + c} \right) $ is a perfect square. So, to mould the given trinomial into a perfect square, we make the expression resemble the square of a binomial term as $ {\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right) $ .

Complete step by step solution:
If a trinomial is a perfect square, it is of the form $ \left( {{a^2} + 2ab + {b^2}} \right) $ which can be condensed as $ {\left( {a + b} \right)^2} $ .
In the given question, we are given a polynomial $ \left( {{x^2} + 9x + c} \right) $ and we have to find the value of c for which the given polynomial becomes a perfect square. In this case, we are given a trinomial $ \left( {{x^2} + 9x + c} \right) $ .
So, we have to convert the given trinomial $ \left( {{x^2} + 9x + c} \right) $ into a whole square. For doing this, we need to make the trinomial resemble form of the whole square of a binomial as $ {\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right) $ .
Hence, $ \left( {{x^2} + 9x + c} \right) $
 $ \Rightarrow \left( {{x^2} + \left( 2 \right)\left( {4.5} \right)x + c} \right) $
So on comparing $ \left( {{a^2} + 2ab + {b^2}} \right) $ and $ \left( {{x^2} + \left( 2 \right)\left( {4.5} \right)x + c} \right) $ .
So, we get the values of $ a = x $ and $ b = 4.5 $ .
Hence, the value of c must be $ {\left( {4.5} \right)^2} $ .
Further doing the calculation, we get,
 $ \Rightarrow c = {\left( {4.5} \right)^2} $
 $ \Rightarrow c = 20.25 $
So, the value of c such that the given trinomial $ \left( {{x^2} + 9x + c} \right) $ is a perfect square is $ \left( {20.25} \right) $ .
So, the correct answer is “ $ \left( {{x^2} + 9x + c} \right) $ is a perfect square is $ \left( {20.25} \right) $ ”.

Note: We can also cross verify the answer by doing the reverse of the process and finding out whether the square of the binomial term gives us the required trinomial. Square is nothing but multiplying the same number with itself. So we will multiply the bracket with itself. Then each individual term in the first bracket is multiplied with that in the second term. Then if needed any mathematical operations, those will be performed or else we can use the standard and important identities used for expansion. Like those used in squaring or cubing. Those are the algebraic identities.