In the equation ax^2+bx+c=0, one root is the square of the other. Without solving the equation, prove that
c(a-b)^3=a(c-b)^3
To prove that c(a-b)^3 = a(c-b)^3 without solving the equation, we can use the property that if one root of a quadratic equation is the square of the other, their sum and product will have a specific relationship.
Let's assume that one root is x, and the other root is x^2.
Using the sum and product of roots formulas for a quadratic equation:
Sum of roots = -b/a
Product of roots = c/a
In this case, the sum of the roots is x + x^2 and the product of the roots is x * x^2 = x^3.
According to the problem, x^2 is the square of x, so we can write x^2 = x * x.
Now, let's expand the given expressions:
c(a - b)^3 = c(a - b)(a - b)(a - b)
= c(a^2 - 2ab + b^2)(a - b)
= c(a^3 - 3a^2b + 3ab^2 - b^3)
a(c - b)^3 = a(c - b)(c - b)(c - b)
= a(c^2 - 2cb + b^2)(c - b)
= a(c^3 - 3c^2b + 3cb^2 - b^3)
Now, we need to show that c(a - b)^3 = a(c - b)^3.
From the expanded expressions above, we can observe that all terms before -b^3 are the same in both expressions. Now, it remains to show that -b^3 is the same as -b^3.
Since we initially assumed that one root is x and the other root is x^2, we take -b = x + x^2.
Now, if we substitute this into the expressions for c(a - b)^3 and a(c - b)^3, the -b^3 terms will cancel out.
Therefore, c(a - b)^3 = a(c - b)^3 even without solving the quadratic equation.