Question

How do you factor \[{{x}^{2}}-28x+196\]?

Answer 1

Hint: Use the middle term split method to factorize \[{{x}^{2}}-28x+196\]. Split -28x into two terms in such a way that their sum equals -28x and product equals \[196{{x}^{2}}\]. To do this, write the prime factors of 196 and group then in such a way that the conditions get satisfied. Finally, take the common terms together and write \[{{x}^{2}}-28x+196\] as a product of two terms given as (x – m) (x – n). Here, ‘m’ and ‘n’ are called zeroes of the polynomial.

Complete step by step solution:
Here, we have been asked to factorize the quadratic polynomial: \[{{x}^{2}}-28x+196\].
Let us use the middle term split method for the factorization. According to this method we have to split the middle term which is -28x into two terms such that their sum is -28x and product is \[196{{x}^{2}}\]. To do this, first we need to find all the prime factors of 196.
We know that 196 can be written as: - \[196=2\times 2\times 7\times 7\] as the product of its primes. Now, we have to group these factors such that the conditions of the middle term split method are satisfied. So, we have,
(i) $\left( -14x \right)+\left( -14x \right)=-28x$
(ii) \[\left( -14x \right)\times \left( -14x \right)=196{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow {{x}^{2}}-28x+196={{x}^{2}}-14x-14x+196 \\
 & \Rightarrow {{x}^{2}}-28x+196=x\left( x-14 \right)-14\left( x-14 \right) \\
\end{align}\]
Taking (x – 14) common in the R.H.S, we have,
\[\begin{align}
  & \Rightarrow {{x}^{2}}-28x+196=\left( x-14 \right)\left( x-14 \right) \\
 & \Rightarrow {{x}^{2}}-28x+196={{\left( x-14 \right)}^{2}} \\
\end{align}\]
Hence, \[{{\left( x-14 \right)}^{2}}\] is the factored form of the given quadratic polynomial.

Note: Here, as you can see we have both the factors of the quadratic polynomial equal to (x – 14), this is because the given expression is of the form of algebraic expression \[{{a}^{2}}-2ab+{{b}^{2}}\], where a = x and b = 14, whose factored form is given directly using the identity \[{{a}^{2}}-2ab+{{b}^{2}}={{\left( a-b \right)}^{2}}\]. So, this can also be a method to factorize the given quadratic polynomial.