How to find the value of a and the value of b?

${x}^{2} - 16 x + a = {\left(x + b\right)}^{2}$

Mar 13, 2017

$a = 64$ and $b = - 8$

Explanation:

This appears to be a way of finding a number $a$, which when added to ${x}^{2} - 16 x$ results in a square of form ${\left(x + b\right)}^{2}$

We can write ${x}^{2} - 16 x + a = {\left(x + b\right)}^{2}$ as

${x}^{2} - 16 x + a = {x}^{2} + 2 b x + {b}^{2}$

Now comparing coefficients of similar terms

$2 b = - 16$ or $b = - 8$

and $a = {b}^{2} = {\left(- 8\right)}^{2} = 64$

Mar 13, 2017

$a = 64 \text{ }$ and $\text{ } b = - 8$

Explanation:

Given:

${x}^{2} - 16 x + a = {\left(x + b\right)}^{2}$

$\textcolor{w h i t e}{{x}^{2} - 16 x + a} = {x}^{2} + 2 b x + {b}^{2}$

Equating the coefficients of $x$, we find:

$- 16 = 2 b$

Hence:

$b = - 8$

Then:

${b}^{2} = {\left(- 8\right)}^{2} = 64$

So equating the constant terms, we find:

$a = 64$