Question

If $a + b = 7$and $ab = 10$, then find $a - b$.
${\text{A}}{\text{. }} \pm {\text{2}}$
${\text{B}}{\text{. }} \pm 7$
${\text{C}}{\text{. }} \pm 1$
${\text{D}}{\text{. }} \pm 3$

Answer 1

Hint: Given equation $a + b = 7$, square both sides of this equation and then find the value of ${a^2} + {b^2}$ and then using this find the value of $a - b$.

Complete step-by-step answer:

We have been given $a + b = 7$ and $ab = 10$.
Now using $a + b = 7$ and squaring both sides we get-
$
  {(a + b)^2} = {7^2} \\
   \Rightarrow {a^2} + {b^2} + 2ab = 49 - (1) \\
 $
Now have $ab = 10$, putting in equation (1), we get-
$
   \Rightarrow {a^2} + {b^2} + 2(10) = 49 \\
   \Rightarrow {a^2} + {b^2} = 49 - 20 = 29 \\
 $
Now, we have to find $a - b$, squaring both sides we get-
${(a - b)^2} = {a^2} + {b^2} - 2ab - (2)$
Putting the value of ${a^2} + {b^2}$ in the equation (2), we get-
$
  {(a - b)^2} = 29 - 2(10) \\
   \Rightarrow {(a - b)^2} = 9 \\
   \Rightarrow (a - b) = \pm 3 \\
$
Hence, the value of $a - b$ is $ \pm 3$.
Therefore, the correct option is ${\text{D}}{\text{. }} \pm 3$.

Note: Whenever such types of questions appear, then write the things given in the equation and then using the equation $a + b = 7$ and then squaring it to find the value of ${a^2} + {b^2}$. Then, do the square of (a – b), and then we will get ${(a - b)^2} = {a^2} + {b^2} - 2ab$, put the value of ${a^2} + {b^2}$ and value of ab as given in the question to find (a – b).