Question

Given ${\text{}}{a^2} + {b^2} = 41{\text{ and }}ab = 4,{\text{ then find }} the\, value\, of\, a - b.$
A.$\sqrt {35} $
B.$6$
C.$\sqrt {33} $
D.$7$

Answer 1

Hint: We have given the value of ${a^2} + {b^2}{\text{ and }}ab$ and we have to calculate the value of $a - b$. We do this by using the mathematical identities of algebra. These are ${\left( {a + b} \right)^2}{\text{ and }}{\left( {a - b} \right)^2}$. The expand form ${\left( {a + b} \right)^2}$ of is ${a^2} + {b^2} + 2ab$ and expand form of ${\left( {a - b} \right)^2}$ is ${a^2} + {b^2} - 2ab$. We have given ${a^2} + {b^2}$ we add we put these values in ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$.
After solving we get the value of $a - b$.

Complete step-by-step answer:
We have the values of
${a^2} + {b^2}{\text{ and }}ab$
Value of ${a^2} + {b^2}{\text{ is }}41$
Value of $ab{\text{ is }}4$
We know that
${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
Putting these values of ${a^2} - {b^2}{\text{ and }}ab$ in this identity.
We get
${\left( {a - b} \right)^2} = 41 - 2 \times 4$
$ = 41 - 8{\text{ }} \Rightarrow {\text{ }}33$
${\left( {a - b} \right)^2} = 33$
Taking square root on both sides we get
$\Rightarrow$ $\sqrt {{{\left( {a - b} \right)}^2}} {\text{ = }}\sqrt {33} $
Square root, cancels square from Left hand Side
$\Rightarrow$ ${\text{ }}a - b{\text{ = }}\sqrt {33} $
Option (C) is correct.

Note: Some expressions in mathematics are made up of some variables and constants with algebraic operations like addition, multiplication, subtraction. They are called algebraic expressions. The constants are called coefficients of the variable. The algebraic expressions are also called algebraic equations
Example: $3x + 4 = 0$.
Here $x$ is a variable whose values are unknown. This can take any value
$3$ is the coefficient of $x$ as it is constant value so it is well defined.
This whole expression has two terms so it is called a binomial expression.
Monomial expression has only one term. In general the expressions which have more than one term are called polynomial. Apart from these expressions the algebraic expression can also be classified into two additional types: Numeric and variable expression.