Question

Find the value of \[a\] and \[b\] in \[{{x}^{2}}-16x+a={{(x+b)}^{2}}\]

Answer 1

Hint: In this type of question when two equations of the same degree are equal given then we just need to expand the expressions on both sides and then compare the coefficients of each term and compare constants to know the required values asked in the question.

Complete step by step answer:
To solve the above question first we need to know the basic methods to solve the equations and we also need to know the basic identities of algebra.
An algebraic equation can be defined as a mathematical statement in which two expressions are set equal to each other. In simple words, equations mean equality i.e. the equal sign. That’s what equations are all about- “equating one quantity with another”
Equations are like a balance scale. If you’ve seen a balance scale, you would know that an equal amount of weight has to be placed on either side for the scale to be considered “balanced”. If we add some weight to just one side, the scale will tip on one side and the two sides are no longer in balance. Equations follow the same logic. Whatever is on one side of the equal sign must have exactly the same value on the other side else it becomes an inequality.
The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and solving different polynomials. You have already learned about a few of them in the junior grades here, we will recall them and introduce you to some more standard algebraic identities, along with examples.
Some of the basic identities are as below:
\[1)\] \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
\[2)\] \[{{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]
\[3)\] \[{{a}^{2}}-{{b}^{2}}=(a+b)(a-b)\]
And we have many more identities but above are most commonly used.
Now to solve when two expressions are equivalent then we should try to compare the equations this way we will find the required values asked in the question above.
Now given that,
  \[{{x}^{2}}-16x+a={{(x+b)}^{2}}\]
On expanding R.H.S by using identity \[(1)\] from above identities we get,
\[\Rightarrow {{x}^{2}}-16x+a={{x}^{2}}+2bx+{{b}^{2}}\]
Now on comparing L.H.S and R.H.S coefficients we have,
\[-16=2b\]
\[\Rightarrow b=\dfrac{-16}{2}\]
\[\Rightarrow b=-8\]
\[b=-8\]
Now comparing other coefficient we have,
\[a={{(b)}^{2}}\]
On putting value of \[b=-8\] we get,
\[\begin{align}
  & \Rightarrow a={{(-8)}^{2}} \\
 & \Rightarrow a=64 \\
 & a=64 \\
\end{align}\]
Hence, as our final answer we got the values \[a=64\] and \[b=-8\] .

Note: We regularly see people using Algebra in many parts of everyday life; for instance, it is utilized in our morning schedule each day to measure the time you will spend in the shower, making breakfast, or driving to work. And it is used to solve so many real world problems.