Question

Using $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ , find
(i) $103 \times 104$
(ii) $5.1 \times 5.2$
(iii) $103 \times 98$
(iv) $9.7 \times 9.8$

Answer 1

Hint: In this question we just need to use the equation and we need to find the values of $a$, $b$ and $x$ from the question and now we need to put this value in the equation which is given to us in the question. After putting the values in the equation we just need to simplify the equation.

Complete step-by-step solution:
(i) $103 \times 104$
We can write $103 \times 104$ as $\left( {100 + 3} \right)\left( {100 + 4} \right)$ and then compare it with $\left( {x + a} \right)\left( {x + b} \right)$. Therefore, we get $x = 100$, $a = 3$ and $b = 4$. Now put all this value in the equation below:
$\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$.
Put the values of $x$ , $a$ and $b$ in the above equation:
$ \Rightarrow \left( {100 + 3} \right)\left( {100 + 4} \right) = {100^2} + \left( {3 + 4} \right)100 + \left( 3 \right)\left( 4 \right)$
Now, we have to solve the above equation:
$
   = 10000 + \left( 7 \right)100 + 12 \\
   = 10000 + 700 + 12 = 10712
 $
Therefore, if we use the equation $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ then also we will get $103 \times 104$ is $10712$
Hence, the answer is $10712$.
(ii) $5.1 \times 5.2$
We can write $5.1 \times 5.2$ as $\left( {5 + 0.1} \right)\left( {5 + 0.2} \right)$ and then compare it with $\left( {x + a} \right)\left( {x + b} \right)$. Therefore, we get $x = 5$, $a = 0.1$ and $b = 0.2$. Now put all this value in the equation below:
$\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$.
Put the values of $x$ , $a$ and $b$ in the above equation:
$ \Rightarrow \left( {5 + 0.1} \right)\left( {5 + 0.2} \right) = {5^2} + \left( {0.1 + 0.2} \right)5 + \left( {0.1} \right)\left( {0.2} \right)$
Now, we have to solve the above equation:
$
   = 25 + (0.3)5 + 0.02 \\
   = 25 + 1.5 + 0.02 = 26.52
 $
Therefore, if we use the equation $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ then also we will get $5.1 \times 5.2$ is $26.52$
Hence, the answer is $26.52$.
(iii) $103 \times 98$
We can write $103 \times 98$ as $\left( {100 + 3} \right)\left( {100 - 2} \right)$ and then compare it with $\left( {x + a} \right)\left( {x + b} \right)$. Therefore, we get $x = 100$,$a = 3$ and $b = - 2$. Now put all this value in the equation below:
$\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$.
Put the values of $x$ , $a$ and $b$ in the above equation:
$ \Rightarrow \left( {100 + 3} \right)\left( {100 + ( - 2)} \right) = {100^2} + \left( {3 + ( - 2)} \right)100 + \left( 3 \right)\left( { - 2} \right)$
Now, we have to solve the above equation:
$
   = 10000 + (1)100 - 6 \\
   = 10000 + 100 - 6 = 10094
 $
Therefore, if we use the equation $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ then also we will get $103 \times 98$ is $10094$
Hence, the answer is $10094$.
(iv) $9.7 \times 9.8$
We can write $9.7 \times 9.8$ as $\left( {10 - 0.3} \right)\left( {10 - 0.2} \right)$ and then compare it with $\left( {x + a} \right)\left( {x + b} \right)$. Therefore, we get $x = 10$, $a = - 0.3$ and$b = - 0.2$. Now put all this value in the equation below:
$\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$
Put the values of $x$ , $a$ and $b$ in the above equation:
$ \Rightarrow \left( {10 + ( - 0.3)} \right)\left( {10 + ( - 0.2)} \right) = {10^2} + \left( {( - 0.3) + ( - 0.2)} \right)10 + \left( { - 0.3} \right)\left( { - 0.2} \right)$
Now, we have to solve the above equation:
\[
   = 100 + ( - 0.3 - 0.2)10 + 0.06 \\
   = 100 - 5 + 0.06 = 95.06
 \]
Therefore, if we use the equation $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ then also we will get $9.7 \times 9.8$ is $95.06$
Hence, the answer is $95.06$.

Note: In this question just be careful when choosing the value of $a$, $b$ and $x$ because if you have chosen the wrong value of $a$, $b$ and $x$. It is going to give you incorrect answers. So always put the values with their sign i.e. If values are with negative sign put the values with negative sign.