if a, b,c are real numbers and not all equal, prove that the equation
(c-a)x^2 -2(a-b)x + (b+c) = 0 has unequal real roots
To prove that the equation
(c-a)x^2 - 2(a-b)x + (b+c) = 0
has unequal real roots, we can use the discriminant of the quadratic equation. The discriminant can provide information about the nature of the solutions (roots) of a quadratic equation.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by:
D = b^2 - 4ac
For the given equation, the coefficients are:
a = (c - a)
b = -2(a - b)
c = (b + c)
Now, let's substitute these values into the discriminant formula:
D = (-2(a - b))^2 - 4(c - a)(b + c)
Expanding and simplifying:
D = 4(a - b)^2 - 4(c - a)(b + c)
Further simplifying:
D = 4(a^2 - 2ab + b^2) - 4(bc - ac - ab - ac)
Combining like terms:
D = 4a^2 - 8ab + 4b^2 - 4bc + 4ac + 4ab + 4ac
D = 4a^2 - 4ab + 4b^2 - 4bc + 8ac
Now, we need to analyze the discriminant to determine the nature of the roots:
If D > 0, then the equation has two unequal real roots.
If D = 0, then the equation has two equal real roots.
If D < 0, then the equation has imaginary roots.
Let's consider each case:
Case 1: D > 0
To prove that the equation has two unequal real roots, we need to show that the discriminant is greater than 0.
4a^2 - 4ab + 4b^2 - 4bc + 8ac > 0
Dividing all terms by 4:
a^2 - ab + b^2 - bc + 2ac > 0
Factoring out common terms:
(a - b)^2 + (a - c)(b - c) > 0
Since a, b, and c are real numbers and not all equal, at least one of the factor expressions must be nonzero. Therefore, the sum of these expressions will be greater than zero, proving that D > 0 and the equation has two unequal real roots.
Case 2: D = 0
If D = 0, then the equation has two equal real roots. However, since we are asked to prove that the roots are unequal, this case is not applicable.
Case 3: D < 0
If D < 0, then the equation has imaginary roots. However, we are asked to prove that the roots are real, so this case is also not applicable.
Therefore, by proving that D > 0, we have shown that the equation has two unequal real roots.