if alpha and beta are the zeroes of the polynomial 2x^2-7x+1 find the value of 1/aplha^2+1/beta^2
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EXPLANATION.
α,β are the zeroes of the polynomial.
⇒ 2x² - 7x + 1.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ α + β = -(-7)/2 = 7/2.
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ αβ = 1/2.
To find :
⇒ 1/α² + 1/β².
Factorizes the equation, we get.
⇒ β² + α²/α²β².
As we know that,
Formula of :
⇒ x² + y² = (x + y)² - 2xy.
Using this formula in equation, we get.
⇒ [(α + β)² - 2αβ]/(αβ)².
Put the value in the equation, we get.
⇒ [(7/2)² - 2(1/2)]/(1/2)².
⇒ [49/4 - 1]/1/4.
⇒ [49 - 4/4]/1/4.
⇒ [45/4/1/4].
⇒ 45/4 x 4/1.
⇒ 45.
⇒ 1/α² + 1/β² = 45.
MORE INFORMATION.
Conjugate roots.
(1) = D < 0.
One roots = α + iβ.
Other roots = α - iβ.
(2) = D > 0.
One roots = α + √β.
Other roots = α - √β.
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