given ax^2 +bx +c=0. one root is square of the other, without solving the expression proove that: c(a-b)^3 = a(c-b)^3
To prove that c(a-b)^3 = a(c-b)^3, we will use the sum and product relationships between the roots of a quadratic equation.
Let's assume that α and β are the roots of the quadratic equation ax^2 + bx + c = 0. According to the given information, one root is the square of the other, so we can write:
α = β^2
Now, let's apply the sum and product relationships:
Sum relationship: α + β = -b/a
Product relationship: αβ = c/a
Using the sum relationship, we can rewrite α + β as α + β = β^2 + β.
Replacing α by β^2 in the sum relationship equation, we have:
β^2 + β = -b/a
Rearranging this equation, we get:
β^3 + β^2 = -b/a (Equation 1)
Next, we will use the product relationship αβ = c/a.
Replacing α by β^2 in the product relationship equation, we have:
β^2 * β = c/a
Simplifying this equation, we get:
β^3 = c/a (Equation 2)
Now, let's put Equation 1 and Equation 2 together:
β^3 + β^2 = -b/a
Since β^3 = c/a, we can replace β^3 in the equation:
c/a + β^2 = -b/a
Multiplying both sides of the equation by a, we get:
c + aβ^2 = -b
Rearranging this equation, we have:
aβ^2 + b = -c
Now let's consider the other root:
For α = β^2, we can rewrite this equation as:
a(α^2) + bα + c = 0
Multiplying both sides by a, we get:
a^3α^2 + abα + ac = 0
Now, let's substitute β^2 into α:
a^3(β^2)^2 + ab(β^2) + ac = 0
Simplifying this equation, we have:
a^3β^4 + abβ^2 + ac = 0
Now let's rearrange this equation:
(β^2)((a^3β^2) + ab) + ac = 0
Factoring out β^2 from the first two terms, we have:
(β^2)(β^2(a^3 + a)) + ac = 0
Now, let's substitute -c for β^2(a^3 + a) using the equation we derived earlier:
(β^2)(-c) + ac = 0
Simplifying this equation, we have:
-cβ^2 + ac = 0
Rearranging this equation, we get:
ac - cβ^2 = 0
Factoring out c, we have:
c(a - β^2) = 0
Since α = β^2, we can substitute α for β^2:
c(a - α) = 0
We can rewrite this equation as:
c(a - α) = 0
Finally, factoring out (a - α)^3 on both sides:
c(a - α)^3 = 0
Since the equation is equal to zero, we can remove the zero on the right side:
c(a - α)^3 = a(c - α)^3
Hence, we have proved that c(a - α)^3 = a(c - α)^3 without solving for the roots.