if alpha beeta are roots of ax²+bx+c find the value of1/alpha²+1/beeta²
Answers
EXPLANATION.
α, β are the roots of the quadratic equation.
⇒ f(x) = ax² + bx + c.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a. - - - - - (1).
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a. - - - - - (2).
To find :
⇒ 1/α² + 1/β².
⇒ β² + α²/α²β².
As we know that,
Formula of :
⇒ (x² + y²) = (x + y)² - 2xy.
Using this formula in the equation, we get.
⇒ 1/α² + 1/β² = [β² + α²]/(α²β²).
⇒ 1/α² + 1/β² = [(α + β)² - 2αβ]/(αβ)².
Put the values in the equation, we get.
⇒ 1/α² + 1/β² = [(-b/a)² - 2(c/a)]/(c/a)².
⇒ 1/α² + 1/β² = [b²/a² - 2c/a]/(c/a)².
⇒ 1/α² + 1/β² = [b² - 2ac/a²]/(c²/a²).
⇒ 1/α² + 1/β² = (b² - 2ac)/a² x a²/c².
⇒ 1/α² + 1/β² = (b² - 2ac)/c².
MORE INFORMATION.
Nature of the roots of the quadratic expression.
(1) = Real and unequal, 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.
Answer:
EXPLANATION.
α, β are the roots of the quadratic equation.
⇒ f(x) = ax² + bx + c.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a. - - - - - (1).
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a. - - - - - (2).
To find :
⇒ 1/α² + 1/β².
⇒ β² + α²/α²β².
As we know that,
Formula of :
⇒ (x² + y²) = (x + y)² - 2xy.
Using this formula in the equation, we get.
⇒ 1/α² + 1/β² = [β² + α²]/(α²β²).
⇒ 1/α² + 1/β² = [(α + β)² - 2αβ]/(αβ)².
Put the values in the equation, we get.
⇒ 1/α² + 1/β² = [(-b/a)² - 2(c/a)]/(c/a)².
⇒ 1/α² + 1/β² = [b²/a² - 2c/a]/(c/a)².
⇒ 1/α² + 1/β² = [b² - 2ac/a²]/(c²/a²).
⇒ 1/α² + 1/β² = (b² - 2ac)/a² x a²/c².
⇒ 1/α² + 1/β² = (b² - 2ac)/c².