If the ratio of the roots of the quadratic equation ax^2 + bx +c =0 is m:n
prove that mnb^2=(m+n)^2 ac
sum: m + n = -b/a
product: mn = c/a
substitute. when proving, you only manipulate/solve one side.
mnb^2=(m+n)^2 ac
(c/a)(b^2) =? (ac)(m+n)^2
(c/a)[a(m + n)]^2 =? (ac)(m+n)^2
(c/a)(a^2)(m+n)^2 =? (ac)(m+n)^2
(ac)(m+n)^2 = (ac)(m+n)^2
this problem is pretty much the same as the one you posted earlier. you should also check Naruto's answer on that post.
Gee! Naruto and Sasuke seem to be the same person!!
To prove the given statement, we need to start with the quadratic equation ax^2 + bx + c = 0, where a, b, and c are constants.
Let's assume the roots of the quadratic equation are α and β.
According to the given information, the ratio of the roots is m:n. This implies that α:β = m:n.
Since α + β = -b/a (from the sum of roots formula), we can define α and β as follows:
α = m * k,
β = n * k,
where k is a constant.
Substituting these values into the equation α + β = -b/a, we get:
m * k + n * k = -b/a.
Simplifying, we can factor out k:
k * (m + n) = -b/a.
Now, let's consider the product of the roots, α * β:
α * β = (m * k) * (n * k) = m * n * k^2.
We know that α * β = c/a (from the product of roots formula), so we can write:
m * n * k^2 = c/a.
Multiplying both sides of this equation by b^2, we have:
m * n * k^2 * b^2 = c/a * b^2.
Now, substituting k * (m + n) = -b/a, we get:
m * n * (m + n)^2 * b^2 = c/a * b^2.
Cancelling out the b^2 terms from both sides, we finally get the desired result:
m * n * (m + n)^2 = c * a.
Hence, we have proved that m * n * (m + n)^2 = c * a, which is the statement we needed to prove.