Question

How do you factor $ 64{x^2} + 112x + 49 $ ?

Answer 1

Hint: Factorization is the process of finding which prime numbers can be multiplied together to make the original number. First of all we will observe the given trinomial and then will find out the method to solve. Here, we can observe that the first term and the last term are whole squares and then will check for the middle term to be a perfect whole square.

Complete step-by-step answer:
Take the given expression: $ 64{x^2} + 112x + 49 $
First term: $ 64{x^2} $
The above term can be re-written as:
First term: $ 64{x^2} = {(8x)^2} $
Similarly,
Last term: $ 49 = {7^2} $
Now, the middle term: $ 112x = 2(8x)(7) $
So, combining all the above terms and writing its equivalent expression it gives -
 $ 64{x^2} + 112x + 49 = {(8x)^2} + 2(8x)(7) + {(7)^2} $
By using sum of two terms whole square identity - $ {(a + b)^2} = {a^2} + 2ab + {b^2} $
 $ 64{x^2} + 112x + 49 = {(8x + 7)^2} $
Hence, the factors of the given expression are $ (8x + 7) $ and $ (8x + 7) $
So, the correct answer is “ $ (8x + 7) $ and $ (8x + 7) $ ”.

Note: Always check and observe the given quadratic equation or the trinomial. There are various methods to solve and get the factors. It can be solved by splitting the middle term or using directly the quadratic formula and then placing it. Be good in squares of the terms. Simplify wisely when finding the roots of the equations. Always remember that the middle term in the perfect square should be equal to the product of the first term and the last term and split the middle term accordingly and keep the signs of the terms double check.