There, We presented the Derivation Of Formulae Relating roots and coefficients of a Quadratic Equation. Solved Examples on applying the formulae are given. Problems for practice are also given in Exercise.
Here we present some more Solved Examples and Exercise problems.
Example 1 : The Quadratic Formula
Solved Example 1 of The Quadratic Formula :
If α, β are the roots of x2 - px + q = 0, then find the value of
α3 + β3
(α2⁄β) + (β2⁄α)
(1⁄α3) + (1⁄β3)
(α⁄β) + (β⁄α)
Solution of Example 1 of The Quadratic Formula :
The given equation is x2 - px + q = 0 Comparing this equation with ax2 + bx + c = 0, we get a = 1, b = -p and c = q α, β are the roots of x2 - px + q = 0 We know, Sum of the roots = α + β = -b⁄a = -(-p)⁄1 = p Product of the roots = αβ = c⁄a = q ⁄1 = q
⇒ a3 + b3 = (a + b)3 - 3ab(a + b) Applying this here, we get α3 + β3 = (α + β)3 - 3αβ( α + β) Using the values of α + β (= p) and αβ (= q) here, we get α3 + β3 = ( p)3 - 3q(p) = p3 - 3pq Ans.
(ii) To find the value of (α2⁄β) + (β2⁄α)
Let A = (α2⁄β) + (β2⁄α) Multiplying both sides with αβ, we get αβA = αβ(α2⁄β) + αβ(β2⁄α)= α3 + β3 Using the values of α3 + β3 (= p3 - 3pq) and αβ (= q) here, we get qA = p3 - 3pq ⇒ A = (p3 - 3pq)⁄q⇒ (α2⁄β) + (β2⁄α) = (p3 - 3pq)⁄q. Ans.
(iii) To find the value of (1⁄α3) + (1⁄β3)
Let A = (1⁄α3) + (1⁄β3) Multiplying both sides with α3β3, we get α3β3A = α3β3(1⁄α3) + α3β3 (1⁄β3) (αβ)3A = β3 + α3 = α3 + β3 Using the values of α3 + β3 (= p3 - 3pq) and αβ (= q) here, we get q3A = p3 - 3pq ⇒ A = (p3 - 3pq)⁄q3⇒ (1⁄α3) + (1⁄β3) = (p3 - 3pq)⁄q3. Ans.
(iv) To find the value of (α⁄β) + (β⁄α) Let A = (α⁄β) + (β⁄α) Multiplying both sides with αβ, we get αβA = αβ(α⁄β) + αβ(β⁄α)= α2 + β2 = (α + β)2 - 2αβ Using the values of α + β (= p) and αβ (= q) here, we get qA = (p)2 - 2q ⇒ A = (p2 - 2q)⁄q ⇒ (α⁄β) + (β⁄α) = (p2 - 2q)⁄q. Ans.
Thus all the four problems of The Quadratic Formula are solved.
A root of px2 + qx + r = 0 is thrice the other root.Show that 3q2 = 16pr.
Solution of Example 2 of The Quadratic Formula :
The given equation is px2 + qx + r = 0 Comparing this equation with ax2 + bx + c = 0, we get a = p, b = q and c = r By data one root is thrice the other root. ⇒ If one root is α, then the other root is 3α α + 3α = -b⁄a = -q⁄p ⇒ 4α = -q⁄p ⇒ α = -q⁄4p......(i) α x 3α = c⁄a ⇒ 3α2 = r⁄p......(ii) Using (i) in (ii), we get 3(-q⁄4p)2 = r⁄p ⇒ 3q2⁄16p2 = r⁄p Multiplying both sides with 16p2, we get 3q2 = (16p2)(r⁄p) = 16pr (Proved.)
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Solve the following problems on The Quadratic Formula.
If α, β are the roots of x2 - (k + 1)x + (k2 + k + 1)⁄2 = 0, then show that α2 + β2 = k
For what values of k does the equation (k - 2)x2 + 2(2k - 3)x + (5k - 6) = 0have equal roots? For those values of k, find those equal roots.
For Answers See at the bottom of the Page.
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