Question

If alpha n beta are zeroes of the quadratic polynomial ax^2 +bx+c ,find the value of alpha^2/beta+beta^2/alpha.​

Answer 1

⠀⠀ || ✪ ϙᴜᴇsᴛɪᴏɴ ✪ ||

If α and β are zeroes of the quadratic polynomial ax² + bx + c ,find the value of (α²/β ) + (β²/α).

⠀⠀ || ✪.Solution.✪ ||

Give:-

• polynomial , ax² + bx + c = 0
• If α and β are zeroes of the quadratic polynomial

Find:-

• (α²/β ) + (β²/α)

Explanation:-

we know,

★ Sum of zeroes = [-(coefficient of x)/(coefficient of x²)]

➩ ( α + β ) = -b/a ...........(1)

★ Product of zeroes = [(constant part)/(coefficient of x²)]

➩ α.β = c/a ...................(2)

Now, calculate ,

★ ( α + β )² = ( α² + β² + 2αβ)

keep value in equ(1) and (2)

➩ (-b/a)² = α² + β² + 2×(c/a)

➩ ( α² + β²) = b²/a² + 2c/a

➩ ( α² + β²) = (b²+2ac)/a² ........(3)

Now,

(α²/β ) + (β²/α)

= ( α³ + β³ )/(α.β)

= ( α + β)( α²+ β²-α β)/α.β

keep value by equ(1) , equ(2) and equ(3)

= (-b/a)[ (b²+2ac)/a² - c/a]

= (-b/a)[(b²+2ac-ac)/a²]

= (-b/a) [ (b²-ac)/a²]

= -b(b²-ac)/a³ Ans.