if a and ß are zeroes of x² - 4x + 1,
find the value of 1/a+1/ß-aß
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Answer:
Given: α,β are zeroes of the polynomial x2−4x+1
To find the value of α1+β1−αβ
Sol: From the given quadratic equation,
α+β=4 and αβ=1
Therefore,
α1+β1−αβ=αβα+β−αβ⟹=4−1=3
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EXPLANATION.
α,β are the zeroes of the polynomial,
⇒ F(x) = x² - 4x + 1.
As we know that,
Sum of zeroes of a quadratic equation,
⇒ α + β = -b/a.
⇒ α + β = -(-4)/1 = 4.
Products of zeroes of quadratic equation,
⇒ αβ = c/a.
⇒ αβ = 1/1 = 1.
To find value of,
⇒ 1/α + 1/β - αβ.
⇒ β + α - (αβ)²/αβ.
Put the values in the equation, we get.
⇒ 4 - (1)²/1.
⇒ 4 - 1.
⇒ 3.
MORE INFORMATION.
Nature of the factors of the quadratic expression.
(1) = Real and different, if b² - 4ac > 0.
(2) = Rational and different, if b² - 4ac is a perfect square.
(3) = Real and equal, if b² - 4ac = 0.
(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.
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