Question

The G.M of roots of the equation $ {x^2} - 18x + 9 = 0 $ is
 $ A)3 $
 $ B)4 $
 $ C)2 $
 $ D)1 $

Answer 1

Hint: First, assume a quadratic equation in any form.
Quadratic means at most power two terms, applied by the condition that the sum of the given roots and product of the given roots. In the quadratic equation, the sum of the root is represented as A.M and the product of the root is represented as G.M
After finding the values of the sum of the two terms and the product of the two terms, the required G.M will be the product of the roots in the square root.
Formula used: The general quadratic equation formula is $ a{x^2} - bx + c = 0 $
The sum of the terms is $ \dfrac{{ - b}}{a} $ and the product of the terms is $ \dfrac{c}{a} $

Complete step by step answer:
From the given that we have the equation $ {x^2} - 18x + 9 = 0 $ and which is the quadratic equation (second degree)
Now compare the given equation from the general quadratic equation, we have $ a{x^2} - bx + c = 0 \Rightarrow {x^2} - 18x + 9 = 0 $
We can see that in place of the variables we have the numbers represented which are $ a = 1,b = 18,c = 9 $
Now as per the sum of the root and product of the root of the quadratic equation we have, $ \dfrac{{ - b}}{a} = -\dfrac{{ - 18}}{1} = 18 $ and $ \dfrac{c}{a} = \dfrac{9}{1} = 9 $
Thus, GM is the roots of the product of the given equation and we get, $ \sqrt {\dfrac{c}{a}} = \sqrt 9 $
Since $ 9 $ is the perfect square as it has the values in square and root terms as $ 9 = {3^2} $
Hence, the GM of roots is $ \sqrt {\dfrac{c}{a}} = \sqrt 9 = 3 $

So, the correct answer is “Option A”.

Note: There is no possibility that $ a $ can never be $ 0 $ in the quadratic equation.
Suppose $ a = 0 $ in the given generalized polynomial, then we get $ ax{}^2 + bx + c = 0 \Rightarrow bx + c = 0 $ then this equation is the linear equation, as it has a degree $ 1 $ .
Thus $ a = 0 $ is not possible in the quadratic equation.
Which are quadratic equations being the equations that are often called the second-degree equation, Like the linear equation are called the first-degree equations.
Quad is referred to as the square of the terms.