Question

If the arithmetic mean of two numbers be A and geometric mean be G, then the numbers will be
A) $A \pm \left( {{A^2} - {G^2}} \right)$
B) $\sqrt A \pm \sqrt {\left( {{A^2} - {G^2}} \right)}$
C) $A \pm \sqrt {\left( {A + G} \right)\left( {A - G} \right)}$
D) $\dfrac{{\left( {A \pm \sqrt {\left( {A + G} \right)\left( {A - G} \right)} } \right)}}{2}$

Answer 1

Here the question is related to the arithmetic mean and the geometric mean. Firstly we write the formula for both the arithmetic mean and geometric mean of two numbers. Then we simplify the geometric mean formula and equate it to the formula of arithmetic mean. Then on simplification we can obtain the numbers.

Complete step by step answer:
The Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the number of values in the set. The formula for the arithmetic mean is defined as $A = \dfrac{{{a_1} + {a_2} + ... + {a_n}}}{n}$.
The geometric mean is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. The formula for the geometric mean is defined as $G = {\left( {{a_1}.{a_2}...{a_n}} \right)^{\dfrac{1}{n}}}$
On considering the given question
The arithmetic mean of two numbers be A. Let the two numbers be a and b. Then arithmetic mean is given by
$\Rightarrow A = \dfrac{{a + b}}{2}$-----(1)
The geometric mean of two numbers be G. It is given as
$\Rightarrow G = {\left( {ab} \right)^{\dfrac{1}{2}}}$----(2)
Now we have to determine the form of numbers will be
So on squaring the equation (2) we have
$\Rightarrow {G^2} = \left( {ab} \right)$
On dividing the above equation by a we have
$\Rightarrow \dfrac{{{G^2}}}{a} = b$----(3)
On substituting equation (3) to equation (1) we have
$\Rightarrow A = \dfrac{{a + \dfrac{{{G^2}}}{a}}}{2}$
On simplifying we have
$\Rightarrow A = \dfrac{{\dfrac{{{a^2} + {G^2}}}{a}}}{2}$
$\Rightarrow A = \dfrac{{{a^2} + {G^2}}}{{2a}}$
$\Rightarrow 2aA = {a^2} + {G^2}$
$\Rightarrow {a^2} - 2aA + {G^2} = 0$
The above equation is in the form of quadratic equation. So now we can determine the value of a.
$\Rightarrow a = \dfrac{{ - ( - 2A) \pm \sqrt {{{( - 2A)}^2} - 4(1)({G^2})} }}{2}$
$\Rightarrow a = \dfrac{{2A \pm \sqrt {4{A^2} - 4{G^2}} }}{2}$
Take 4 as common in the square root and take it out from square root.
$\Rightarrow a = \dfrac{{2A \pm 2\sqrt {{A^2} - {G^2}} }}{2}$
Take 2 as a common in the numerator
$\Rightarrow a = \dfrac{{2(A \pm \sqrt {{A^2} - {G^2}} )}}{2}$
$\Rightarrow a = A \pm \sqrt {{A^2} - {G^2}}$
As we know that the algebraic identity ${a^2} - {b^2} = (a + b)(a - b)$
$\Rightarrow a = A \pm \sqrt {(A - G)(A + G)}$
Therefore, option (C) is the correct option.

Note:
The equation is a quadratic equation. The roots can be solved by using the sum product rule. This defines as for the general quadratic equation $a{x^2} + bx + c$, the product of $a{x^2}$ and c is equal to the sum of $bx$ of the equation by using this we can form the equation and by using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ we can determine the roots for the equation.