Question

A quadratic equation whose difference of roots is $4\sqrt 2 $ and sum of the squares of the roots is 64 is given by
A) ${x^2} + 4\sqrt 6 x + 16$
B) ${x^2} + 4\sqrt 6 x - 16$
C) ${x^2} - 4\sqrt 6 x - 16$
D) ${x^2} - 4\sqrt 6 x + 16$

Answer 1

Hint:
With the given difference of the roots and sum of the squares of the roots we can find the product of the roots using the identity ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ and now with the product of the roots and sum of squares of the roots we get the sum of the roots using the identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$. With the sum and product of the roots the equation is given by ${x^2} - (\text{sum})x + \text{product}$.

Complete step by step solution:
We are given that the difference of roots is $4\sqrt 2 $ and sum of the squares of the roots is 64
Now let the roots be a and b
Since the difference of roots is $4\sqrt 2 $
$ \Rightarrow a - b = 4\sqrt 2 $ ……..(1)
We are given the sum of the squares of the roots is 64
$ \Rightarrow {a^2} + {b^2} = 64$ …………(2)
By using the identity ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
Substituting the given values we get
$
   \Rightarrow {\left( {4\sqrt 2 } \right)^2} = 64 - 2ab \\
   \Rightarrow 16*2 = 64 - 2ab \\
   \Rightarrow 32 - 64 = - 2ab \\
   \Rightarrow - 32 = - 2ab \\
   \Rightarrow \dfrac{{32}}{2} = ab \\
   \Rightarrow ab = 16 \\
 $
Hence we get the product of the roots to be 16
Now we need the sum of the roots to find the quadratic equation
So we can use the identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$to find the sum of the roots
$
   \Rightarrow {\left( {a + b} \right)^2} = 64 + 2\left( {16} \right) \\
   \Rightarrow {\left( {a + b} \right)^2} = 64 + 32 \\
   \Rightarrow {\left( {a + b} \right)^2} = 96 \\
   \Rightarrow \left( {a + b} \right) = \sqrt {96} \\
   \Rightarrow \left( {a + b} \right) = \sqrt {16*6} \\
   \Rightarrow a + b = 4\sqrt 6 \\
 $
Hence with the sum and product of the roots we can get the quadratic equation by
$
   \Rightarrow {x^2} - (sum)x + product \\
   \Rightarrow {x^2} - 4\sqrt 6 x + 16 \\
 $

Therefore the correct option is d.

Note:
The general form of the quadratic equation with roots a and b is given by
$
   \Rightarrow (x - a)(x - b) = {x^2} - ax - bx + ab \\
   \Rightarrow (x - a)(x - b) = {x^2} - (a + b)x + ab \\
 $
Hence the quadratic equation is given as ${x^2} - (\text{sum})x + \text{product}$
Quadratic equations are actually used in everyday life, as when calculating areas, determining a product's profit or formulating the speed of an object. Quadratic equations refer to equations with at least one squared variable, with the most standard form being $a{x^2} + bx + c$ .