Question

Simplify:$(987 \times 987) + 987 \times 26 + 169$

Answer 1

Hint: The above problem can be solved using the algebraic identity of $(a + b)^2$ .
${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ (expansion of the identity)
A plus b whole square is equal to the a square plus b square plus two multiplied by a and b.
We will convert the given expression in the form of above algebraic identity and then we will come to the final answer in single digit.

Complete step-by-step solution:
Let's explain the given algebraic identity in more detail.
$ \Rightarrow {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
This comes as;
${\left( {a + b} \right)^2} = {a^2} + {b^2} + ab + ba$
Which becomes;
${a^2} + {b^2} + 2ab$
Now, we will convert the given expression given in the Question in the form of identity;
$ \Rightarrow (987 \times 987) + 987 \times 26 + 169$
We know that 169 is the square of 13 and we can break 26 into $13 \times 2$;
Therefore we can write as;
$ \Rightarrow (987 \times 987) + 987 \times 13 \times 2 + (13 \times 13)$
The above problem has the form ${a^2} + {b^2} + 2ab$
Where, $a = 987,b = 13$
Now we can write it as;
$ \Rightarrow {\left( {987 + 13} \right)^2}$ (We have combined the two terms in the form of identity)
On adding the two given values inside the bracket we have;
$ \Rightarrow {1000^2}$
On squaring the given value
$ \Rightarrow 1000000$

The value of $(987 \times 987) + 987 \times 26 + 169$ is equal to 1000000

Note: Apart from mentioned identity another algebraic identity is;
${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ (when the two values are subtracted and squared)
${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ (a squared minus b square is equal to a plus b multiplied by a minus b)
The above question could also be solved using the BODMAS rule. Which means that first we will open the bracket, then the division sign is given preference, after that multiplication, addition then subtraction. Using the BODMAS rule is a lengthy method to use for this question, as the multiplication is to be done for three digit numbers.