Question

If we have an expression \[\dfrac{{2\sqrt 7 + 3\sqrt 5 }}{{\sqrt 7 + \sqrt 5 }} = P\sqrt {35} + Q\], then what is the value of \[2P + Q\]

Answer 1

Hint: Here we are asked to find the value of the expression \[2P + Q\]. We are also given an expression, using that we will find the required value of the expression. First, we will try to simplify the given expression to get the required expression form to get its value.
Formula Used: Formula that we need to know before solving this problem:
\[\sqrt a \times \sqrt a = a\]
\[\sqrt a \times \sqrt b = \sqrt {ab} \]
\[{\sqrt a ^2} = a\]

Complete step-by-step solution:
It is given that \[\dfrac{{2\sqrt 7 + 3\sqrt 5 }}{{\sqrt 7 + \sqrt 5 }} = P\sqrt {35} + Q\] we aim to find the value of the expression \[2P + Q\].
First, we will do the conjugate multiplication to simplify the given expression.
\[\dfrac{{2\sqrt 7 + 3\sqrt 5 }}{{\sqrt 7 + \sqrt 5 }} \times \dfrac{{\sqrt 7 - \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }} = P\sqrt {35} + Q\]
When we multiply the numerator and denominator separately on the left-hand side we get
\[\Rightarrow \dfrac{{\left( {2\sqrt 7 \times \sqrt 7 } \right) + \left( {3\sqrt 5 \times \sqrt 7 } \right) - \left( {2\sqrt 7 \times \sqrt 5 } \right) - \left( {3\sqrt 5 \times \sqrt 5 } \right)}}{{{{\sqrt 7 }^2} - {{\sqrt 5 }^2}}} = P\sqrt {35} + Q\]
Now using the formula \[\sqrt a \times \sqrt a = a\] we get
\[\Rightarrow \dfrac{{\left( {2(7)} \right) + \left( {3\sqrt 5 \times \sqrt 7 } \right) - \left( {2\sqrt 7 \times \sqrt 5 } \right) - \left( {3(5)} \right)}}{{{{\sqrt 7 }^2} - {{\sqrt 5 }^2}}} = P\sqrt {35} + Q\]
On simplifying this we get
\[\Rightarrow \dfrac{{14 + \left( {3\sqrt 5 \times \sqrt 7 } \right) - \left( {2\sqrt 7 \times \sqrt 5 } \right) - 15}}{{{{\sqrt 7 }^2} - {{\sqrt 5 }^2}}} = P\sqrt {35} + Q\]
Now let us use the formula \[\sqrt a \times \sqrt b = \sqrt {ab} \] to simplify the above expression
\[\Rightarrow \dfrac{{14 + \left( {3\sqrt {5 \times 7} } \right) - \left( {2\sqrt {7 \times 5} } \right) - 15}}{{{{\sqrt 7 }^2} - {{\sqrt 5 }^2}}} = P\sqrt {35} + Q\]
On further simplification we get
\[\Rightarrow \dfrac{{\left( {3\sqrt {35} } \right) - \left( {2\sqrt {35} } \right) - 1}}{{{{\sqrt 7 }^2} - {{\sqrt 5 }^2}}} = P\sqrt {35} + Q\]
Now using the formula \[{\sqrt a ^2} = a\] we get
\[\Rightarrow \dfrac{{\left( {3\sqrt {35} } \right) - \left( {2\sqrt {35} } \right) - 1}}{{7 - 5}} = P\sqrt {35} + Q\]
Simplifying the above we get
\[\Rightarrow \dfrac{1}{2}\sqrt {35} + \left( { - \dfrac{1}{2}} \right) = P\sqrt {35} + Q\]
On further simplification we get
\[\Rightarrow \dfrac{1}{2}\sqrt {35} - \dfrac{1}{2} = P\sqrt {35} + Q\]
\[\Rightarrow \dfrac{1}{2}\sqrt {35} + \left( { - \dfrac{1}{2}} \right) = P\sqrt {35} + Q\]
On equating the above equation, we get
\[\Rightarrow P = \dfrac{1}{2}\] and \[Q = - \dfrac{1}{2}\]
Thus, we found the value of the terms \[P\] and \[Q\]
Since we are aiming to find the value of the expression \[2P + Q\] , substitute the values of \[P\] and \[Q\] in this expression.
\[\Rightarrow 2P + Q = 2\left( {\dfrac{1}{2}} \right) + \left( { - \dfrac{1}{2}} \right)\]
On simplifying this we get
\[\Rightarrow 2P + Q = \dfrac{1}{2}\]
Thus, we have found the value of the required equation.

Note: To get rid of square roots in our calculation we have used conjugate multiplication. The conjugate of the term \[a + \sqrt b \] is \[a - \sqrt b \]. Conjugate multiplication is nothing but multiplying the conjugate of a denominator to the numerator and also the denominator separately to make the square root into a normal form which will make our calculation easier.