Question

If we have an algebraic expression ${{\left( a-5 \right)}^{2}}+{{\left( b-6 \right)}^{2}}=0$, then find the value of (a+b-3)?
(a) 8
(b) -8
(c) 10
(d) -10

Answer 1

Hint: First, before proceeding for this, we must know the fact that when we perform the square of any term either positive or negative both will give the positive result. Then, we also know that the addition of two positive numbers will always give us a number greater than zero. Then, by applying the above-mentioned condition, we get the values of a and b and then the value of expression also.

Complete step-by-step solution:
In this question, we are supposed to find the value of (a+b-3) when ${{\left( a-5 \right)}^{2}}+{{\left( b-6 \right)}^{2}}=0$.
So, before proceeding with this, we must know the fact that when we perform the square of any term either positive or negative both will give a positive result.
Moreover, we also know that the addition of two positive numbers will always give us a number greater than zero.
So, we can analyze from the given question that in the expression given which is ${{\left( a-5 \right)}^{2}}+{{\left( b-6 \right)}^{2}}=0$, the addition of the two terms gives the result as zero which is only possible when both the terms will be zero.
Now, by applying the above mentioned condition, we get the values of a and b as:
$\begin{align}
  & a-5=0 \\
 & \Rightarrow a=5 \\
\end{align}$ and $\begin{align}
  & b-6=0 \\
 & \Rightarrow b=6 \\
\end{align}$
So, we get the value of a as 5 and b as 6.
Now, we need to calculate the value of the expression $(a+b-3)$ which is asked in the question.
So, by substituting the value of a as 5 and b as 6, we get:
$5+6-3=8$
So, we get the value of the expression $(a+b-3)$ from the condition ${{\left( a-5 \right)}^{2}}+{{\left( b-6 \right)}^{2}}=0$ as 8.
Hence, option (a) is correct.

Note: Now, to solve these types of the questions we need to know some of the basics of the number which tells us about the fact that if the number is squared then it always gives the positive value irrespective of its sign. So, this is the only condition which helps us to solve the above problem and moreover the addition of two numbers is always greater than zero.