Question

I alpha and bitta are the zeros o the polynomial 6xsquare+x-2, ind the value o alpha/bitta+bitta/alpha

Answer 1

Step-by-step explanation:

Given -

• α and β are zeroes of polynomial p(x) = 6x² + x - 2

To Find -

• Value of α/β + β/α

→ α² + β²/αβ

Method 1 :-

Now,

→ 6x² + x - 2

By middle term splitt :-

→ 6x² + 4x - 3x - 2

→ 2x(3x + 2) - 1(3x + 2)

→ (2x - 1)(3x + 2)

Zeroes are -

→ 2x - 1 = 0 and 3x + 2 = 0

→ x = 1/2 and x = -2/3

Hence,

The value of α = 1/2 and β = -2/3

Then,

The value of α/β + β/α is

→ 1/2÷-2/3 + -2/3÷1/2

→ -3/4 + (-4/3)

→ -3/4 - 4/3

→ -9-16/12

→ -25/12

Hence,

The value of α/β + β/α is -25/12

Method 2 :-

As we know that :-

• α + β = -b/a

→ α + β = -1/6

Now, Squaring both sides :-

→ (α + β)² = (-1/6)²

→ α² + β² + 2αβ = 1/36

→ α² + β² + 2(-1/3) = 1/36

→ α² + β² - 2/3 = 1/36

→ α² + β² = 1/36 + 2/3

→ α² + β² = 1 + 24/36

→ α² + β² = 25/36

And

• αβ = c/a

→ αβ = -2/6

→ αβ = -1/3

Then,

The value of α² + β²/αβ is

→ 25/36 ÷ -1/3

→ 25/36 × -3

→ -25/12

Hence,

The value of α/β + β/α or α² + β²/αβ is -25/12.

Answer 2

$\large{\boxed{\underline{\underline{\bf{\red{Answer:-}}}}}}$

$\implies$ x = 1/2

$\implies$ x = -2/3

$\large\underline\mathrm{The \: value \: of \: α \: = \: 1/2 \: or \: β \: = \: -2/3}$

$\large\underline\mathrm{Given:-}$

• β and α are zeroes of polynomial p(x) = 6x² + x - 2

$\large\underline\mathrm{To \: find}$

• value of α/β + β/α

$\implies$ α² + β²/αβ

$\large\underline\mathrm{Now,}$

$\implies$ 6x² + x - 2

$\implies$ 6x² + 4x - 3x - 2

$\implies$ 2x(3x + 2) - 1(3x + 2)

$\implies$ (2x - 1)(3x + 2)

$\implies$ 2x - 1= 0

$\implies$ 2x = 1

$\implies$ x = 1/2

$\implies$ 3x + 2 = 0

$\implies$ 3x = -2

$\implies$ x = -2/3

$\large\underline\mathrm{Hence,}$

The value of α = 1/2 or β = -2/3

$\large\underline\mathrm{Then,}$

The value of α/β + β/α is

$\implies$ 1/2 ÷ -2/3 + -2/3 ÷ 1/2

$\implies$ -3/4 + (-4/3)

$\implies$ -3/4 - 4/3

$\implies$ -9 - 16/12

$\implies$ -25/12

$\large\underline\mathrm{Hence,}$

The value of α/β + β/α is -25/12

$\implies$ α + β = -b/a

$\implies$ α + β = -1/6

$\large\underline\mathrm{Both \: sides \: of \: square \: : \: -}$

$\implies$ (α + β)² = (-1/6)²

$\implies$ α² + β² + 2αβ = 1/36

$\implies$ α² + β² + 2(-1/3) = 1/36

$\implies$ α² + β² - 2/3) = 1/36

$\implies$ α² + β² = 2/3 + 1/36

$\implies$ α² + β² = 1 + 24/36

$\implies$ α² + β² = 25/36

$\large\underline\mathrm{And}$

$\implies$ αβ = c/a

$\implies$ αβ = -2/6

$\implies$ αβ = -1/3

$\large\underline\mathrm{Then,}$

The value of α² + β²/βα is

$\implies$ 25/36 ÷ -1/3

$\implies$ 25/36 × -3

$\implies$ 25/12

$\large\underline\mathrm{Hence,}$

The value of α/β + β/α or α² + β²/βα is -25/12

$\large\underline\mathrm{Hope \: it \: helps \: you \: plz \: mark \: me \: brainlist}$