If α and β are the zeroes of the quadratic polynomial x2 –5x + 6, find the value of α4β2 + α2β4.
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EXPLANATION.
α and β are the zeroes of the quadratic polynomial.
⇒ f(x) = x² - 5x + 6.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ α + β = -(-5)/1 = 5.
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ αβ = 6/1 = 6.
To find :
⇒ α⁴β² + a²β⁴.
⇒ α⁴β² + a²β⁴ = α²β²[a² + β²].
As we know that,
Formula of :
⇒ x² + y² = (x + y)² - 2xy.
Using this formula in equation, we get.
⇒ α⁴β² + a²β⁴ = (αβ)²[(α + β)² - 2αβ].
⇒ α⁴β² + a²β⁴ = (6)²[(5)² - 2(6)].
⇒ α⁴β² + a²β⁴ = 36[25 - 12].
⇒ α⁴β² + a²β⁴ = 36[13].
⇒ α⁴β² + a²β⁴ = 468.
MORE INFORMATION.
Nature of the roots of the quadratic expression.
(1) = Real and unequal, if b² - 4ac > 0.
(2) = Rational and equal, 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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