Question

If one of the roots of equation:

ax²+bx+c=0

be the square of other ,show that

b³+a²c+ac²=3abc

Answer 1

GIVEN: one root of quadratic polynomial

ax²+bx+c is square of the other.

So let's assume its roots are β & β²

TO FIND THE VALUE OF: b^3+ac²+a²c

As we know

(1) sum of the roots of a quadratic polynomial = -b/a

(2) product of its roots = c/a

So, β+β² = -b/a…………(1)

& β x β² = c/a

Or. β^3 = c/a……………..(2)

Now, as we have to find the value of an expression, in which we have b^3

So by cubing ……..(1)

(β+β²)^3 = (-b)^3 / a^3

= β^3 + β^6 + 3β^4 + 3β^5 = -b^3 / a^3

Now we convert the above equation in the form of (β+β²) & β^3 form.

= β^3 (1+β^3+3β+3β² ) = -b^3/a^3

= β^3 ( 1+β^3+3(β+β²) = -b^3/a^3……..(3)

So now we can put up the value in the above equation by (1)&(2)

We get, c/a{1+c/a+3(-b/a)} = -b^3/a^3

=> c/a + c²/a² - 3bc/a² = -b^3/a^3

=> (ac+c²-3bc) /a² = -b^3/a^3

Now multiplying a^3 on both sides

a²c+ac²-3abc = -b^3

Or, b^3 + ac² + a²c = 3abc Ans