Question

If \[{a^2} + {b^2} = 29\] and \[ab = 10\], then find \[a - b\].
A.10
B.3
C.9
D.19

Answer 1

Hint: Here we can apply the algebraic formula as \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] . As we are given all the values required, first we calculate \[{(a - b)^2}\] and then take square root to get the desired answer.

Complete step-by-step answer:
As the given algebraic values are, \[{a^2} + {b^2} = 29\] and \[ab = 10\] ,
Let us use the algebraic formula of \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] .
As we know the values of \[{a^2} + {b^2} = 29\] and \[ab = 10\]
Now, on substituting the values in the above equation, we get,
 \[
  {(a - b)^2} = {a^2} + {b^2} - 2ab \\
   \Rightarrow {(a - b)^2} = (29) - 2(10) \\
 \]
On simplification we get,
 \[ \Rightarrow {(a - b)^2} = 9\]
On taking positive square root we get,
 \[ \Rightarrow (a - b) = 3\]
Hence, it is clear from the above given solution that option (B) is our required correct answer.

Additional information :
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
A polynomial equation is an equation that has multiple terms made up of numbers and variables.
The degree tells us how many roots can be found in a polynomial equation.
For example, if the highest exponent is 3, then the equation has three roots.

Note: The above equation can also be proved by the method given below as shown,
 \[
  (a - b)(a - b) = {a^2} + {b^2} - ab - ab \\
   \Rightarrow (a - b)(a - b) = {a^2} + {b^2} - 2ab \\
 \]