Question

How do you factor \[{{x}^{2}}+16x+64\] ?

Answer 1

Hint: In this problem, we factorize using what is known as the middle term method. Here, we rewrite the middle term of the quadratic expression as a sum of two numbers, such that the product of these two numbers is same as that of \[ac\] of the general quadratic expression expressed as \[a{{x}^{2}}+bx+c\]. After rewriting, now there are a total of four terms. We then make two linear factors out of these four terms.

Complete step-by-step solution:
The general quadratic expression is expressed as,
\[a{{x}^{2}}+bx+c\]
In this problem, \[a=1\], \[b=16\] and \[c=64\].
We now find two such numbers which satisfy two conditions simultaneously, first condition being the product of those two numbers must be equal to ac and the second condition being the sum of those two numbers must be equal to b.
After a little trial and error, we get two such numbers \[8\] and \[8\] itself. The product \[8\times 8=64\] and their sum \[8+8=16\]. Thus, both the conditions are satisfied. Now, we rewrite the middle term of the quadratic expression as
\[{{x}^{2}}+8x+8x+64\]
Now, we take \[x\] common from the first two terms and \[8\] common from the last two terms and get
\[x\left( x+8 \right)+8\left( x+8 \right)\]
Now, we take \[\left( x+8 \right)\] common from the two terms and get
\[\begin{align}
  & \left( x+8 \right)\left( x+8 \right) \\
 & \Rightarrow {{\left( x+8 \right)}^{2}} \\
\end{align}\]
Thus, the given expression gets factored to \[{{\left( x+8 \right)}^{2}}\].

Note: The given expression can also be solved by the vanishing factor method. We need to find a number by trial and error which is a root of the quadratic expression. Then, \[x\]– (this number) will be a factor of this expression. The other factors can be found out by dividing the expression by this factor.