Show that if roots of eqn (a²-bc) x²+2(b²-ac)x +c²-ab=0 are equal then either b=0 or a³+b³+c³=3abc.

To prove that either b=0 or a³+b³+c³=3abc, let's start by assuming that the roots of the equation are equal.

The quadratic equation given is:

(a² - bc)x² + 2(b² - ac)x + (c² - ab) = 0

For roots to be equal, the discriminant of the equation (b² - 4ac) must be equal to zero.

Let's calculate the discriminant and set it equal to zero:

Discriminant (D) = (2(b² - ac))^2 - 4(a² - bc)(c² - ab)
= 4(b⁴ - 2ab²c + a²c²) - 4(a²c² - b²ac - abc² + ab³)
= 4b⁴ - 8ab²c + 4a²c² - 4a²c² + 4b²ac + 4abc² - 4ab³
= 4b⁴ - 8ab²c + 4b²ac + 4abc² - 4ab³

Since the roots are equal, the discriminant D must be equal to zero. Therefore, we have:

4b⁴ - 8ab²c + 4b²ac + 4abc² - 4ab³ = 0

Simplifying this equation, we get:

4b⁴ - 8ab²c + 4b(b² - ac)c + 4ab(bc - ab) = 0

Now, let's factor out the common terms:

4b(b³ - abc - (b² - ac)c + bc - ab) = 0

Further simplifying, we have:

4b(b³ - abc - b²c + ac² + bc - ab) = 0

Now, we have two cases to consider:

Case 1: b = 0
If b = 0, then the equation is satisfied, and we have one possible condition.

Case 2: b ≠ 0
If b ≠ 0, we can divide the equation by 4b:

b³ - abc - b²c + ac² + bc - ab = 0

Rearranging the terms, we get:

-abc - b²c - ab + ac² + bc + b³ = 0

Now, we can regroup the terms:

(-abc + ac² + b³) + (-ab - b²c + bc) = 0

Factoring out common terms, we have:

ac(c - b) + (-a - bc)(b² - 1) = 0

Now, we have two cases again:

Case 2.1: (c - b) = 0
If c = b, then the equation simplifies to:

ac(c - b) + (-a - bc)(b² - 1) = ac(0) + (-a - ab)(b² - 1)
= 0 + (-a - ab)(b² - 1)

By factoring out (-a - ab), we get:

(-a - ab)(b² - 1) = 0

Since (-a - ab) cannot be zero, it is clear that (b² - 1) must be zero:

b² - 1 = 0

Solving this equation, we find:

b² = 1

Taking the square root of both sides, we have two possibilities:

b = 1 or b = -1

Case 2.2: (-a - bc) = 0
If -a - bc = 0, then we have:

ac(c - b) + (-a - bc)(b² - 1) = 0

Substituting -a = bc, we get:

ac(c - b) + (bc - bc)(b² - 1) = 0

Simplifying further, we have:

ac(c - b) = 0

This equation holds true when c - b = 0 or ac = 0.

If c = b, then we have one possible condition.

If ac = 0, then either a = 0 or c = 0.

Thus, we have shown that if the roots of the given quadratic equation are equal, either b = 0 or a³ + b³ + c³ = 3abc.

If the roots of a quadratic equation are equal, that means the discriminant equals zero, or

b²-4ac=0 (for ax²+bx+c=0)

So expand the expression and see if you can factorize the result to suit the conditions.