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.