α and β are the zeroes of a quadratic polynomial.
⇒ f(x) = 2x² - 5x - 6.
As we know that,
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = - b/a.
⇒ α + β = - (-5/2).
⇒ α + β = 5/2.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
⇒ αβ = (-6/2) = - 3.
⇒ αβ = - 3.
(α + β) and (αβ) are the zeroes of quadratic polynomial.
Let, we assume that,
⇒ (α + β) + (αβ) = - b/a.
⇒ (5/2) + (-3).
⇒ (5 - 6)/2 = - 1/2.
⇒ (α + β) + (αβ) = -1/2.
⇒ (α + β) x (αβ) = c/a.
⇒ (5/2) x (-3) = - 15/2.
⇒ (α + β) x (αβ) = -15/2.
Formula of quadratic polynomial.
⇒ x² - (α + β)x + αβ.
Formula of quadratic equation whose roots are : (α + β) and (αβ).
⇒ x² - [(α + β) + (αβ)]x + (α + β) x (αβ).
Put the values in the equation, we get.
⇒ x² - (-1/2)x + (-15/2) = 0.
⇒ 2x² + x - 15 = 0.
Quadratic polynomial whose zeroes are : (α + β) and (αβ) = 2x² + x - 15 = 0.
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Answers & Comments
EXPLANATION.
α and β are the zeroes of a quadratic polynomial.
⇒ f(x) = 2x² - 5x - 6.
As we know that,
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = - b/a.
⇒ α + β = - (-5/2).
⇒ α + β = 5/2.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
⇒ αβ = (-6/2) = - 3.
⇒ αβ = - 3.
(α + β) and (αβ) are the zeroes of quadratic polynomial.
As we know that,
Let, we assume that,
Sum of the zeroes of the quadratic polynomial.
⇒ (α + β) + (αβ) = - b/a.
⇒ (5/2) + (-3).
⇒ (5 - 6)/2 = - 1/2.
⇒ (α + β) + (αβ) = -1/2.
Products of the zeroes of the quadratic polynomial.
⇒ (α + β) x (αβ) = c/a.
⇒ (5/2) x (-3) = - 15/2.
⇒ (α + β) x (αβ) = -15/2.
Formula of quadratic polynomial.
⇒ x² - (α + β)x + αβ.
Formula of quadratic equation whose roots are : (α + β) and (αβ).
⇒ x² - [(α + β) + (αβ)]x + (α + β) x (αβ).
Put the values in the equation, we get.
⇒ x² - (-1/2)x + (-15/2) = 0.
⇒ 2x² + x - 15 = 0.
Quadratic polynomial whose zeroes are : (α + β) and (αβ) = 2x² + x - 15 = 0.