Question

if alpha and beta are the zeroes of the polynomial 2x^2-7x+1 find the value of 1/aplha^2+1/beta^2​

Answer 1

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.