Question

The different between the roots of a quadratic equation is 4.If the difference between the cubes of those roots is 208 , then find the quadratic equation.

Answer 1

   ;hope the following answer would help u

Answer 2

Answer:

The required equation is $x^2+8x+12=0$

Step-by-step explanation:

Given : The different between the roots of a quadratic equation is 4.If the difference between the cubes of those roots is 208.

To find : The quadratic equation?

Solution :

Let a and b are the roots of the equation.

The different between the roots of a quadratic equation is 4.

i.e. $a-b=4$ ......(1)

If the difference between the cubes of those roots is 208.

i.e. $a^3-b^3=208$ ......(2)

Cubing both side in equation (1),

$(a-b)^3=4^3$

$a^3-b^3-3ab(a-b)=64$

Substitute the value from eqn.(1) and (2)

$208-3ab(4)=64$

$12ab=208-64$

$12ab=144$

$ab=12$ ......(3)

Product of the zeros is 12.

Squaring both side in equation (1),

$(a-b)^2=4^2$

$a^2+b^2-2ab=16$

Substitute ab=12,

$a^2+b^2-2(12)=16$

$a^2+b^2=16+24$

$a^2+b^2=40$ ......(4)

Using identity,

$(a+b)^2=a^2+b^2+2ab$

Substitute the value from eqn. (3) and (4)

$(a+b)^2=40+2(12)$

$(a+b)^2=40+24$

$(a+b)^2=64$

$a+b=8$

The sum of the zeros is 8.

The quadratic equation form is

$x^2+(a+b)x+(ab)=0$

Substitute the values,

$x^2+(8)x+(12)=0$

The required equation is $x^2+8x+12=0$