Question

How do you solve for the value of \[a\] and \[b\]
\[{x^2} + 16x + a = {\left( {x + b} \right)^2}\]

Answer 1

Hint:
In the given question, we have been given two unknowns in a quadratic equation. We have to find their value. We are going to do that by opening the brackets, solving the value of the expression, comparing the terms on the two sides of the equality and equating the value of the unknowns by comparison.

Formula Used:
We are going to use the formula of whole square,
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]

Complete step by step solution:
The given equation is,
\[{x^2} + 16x + a = {\left( {x + b} \right)^2}\]
Solving the bracket,
\[{x^2} + 16x + a = {x^2} + 2bx + {b^2}\]
Now, we compare the two sides,
\[2b = 16 \Rightarrow b = 8\]
and \[a = {b^2}\]
But we just calculated that \[b = 8\], hence,
\[a = {b^2} = {8^2} = 64\]

Thus, \[a = 64\] and \[b = 8\]

Note:
In the given question we had to find the values of two unknowns used in a quadratic equation. We did that by opening the brackets, solving the value of the expression, comparing the terms on the two sides of the equality and equating the value of the unknowns by comparison.