Question

alpha and beta are zeros of polynomial 4 x square + 4 x minus 1 then find the value of one upon alpha + 1 upon beta​

Answer 1

Given, a quadratic polynomial:

p(x) = 4x² + 4x - 1

First, Let us find the zeroes of p(x) using the quadratic formula:

Comparing the given polynomial with the standard form of quadratic equation excluding equality i.e., ax² + bx + c , we get

• a = 4 , b = 4 , c = -1

⇒ x = (-b ± √{b² - 4ac})/ 2a

⇒ x = (-4 ± √{ 16 + 16 } ) / 8

⇒ x = ( -4 ± 4√2 ) / 8

⇒ x = (±√2 - 1)/ 2

We have,

x = (√2 - 1) / 2 , -(√2 + 1) / 2

So,

• ɑ = (√2 - 1)/2 , β = -(√2 + 1)/2

Now, We need to find the value of 1/ɑ + 1/β

⇒ 1/(√2 - 1)/2 + 1/(-√2 - 1)/2

⇒ 2/(√2 - 1) - 2/(√2 + 1)

⇒ 2√2 + 2 - 2√2 + 2

⇒ 4

Hence, The value of 1/ɑ + 1/β is 4.

Answer 2

$\large\bf{\underline{\underline\red{Given:-}}}$

• α and β are zeroes of the polynomial 4x² + 4x -1.

$\large\bf{\underline{\underline\blue{To\:Find:-}}}$

• $\rm The \: value \: of \: \dfrac{1}{ \alpha } \: and \: \dfrac{1}{ \beta }$

$\large\bf{\underline{\underline\green{Solution:-}}}$

Given polynomial :- 4x² + 4x - 1

Zeroes = α and β

As we know that:-

For a quadratic polynomial ax² + bx + c

$\rm Sum \: of \: zeroes = \dfrac{ - b}{a}$

$\rm Product \: \: of \: zeroes = \dfrac{c}{a}$

Comparing 4x² + 4x - 1 with ax² + bx + c

Here:-

• a = 4

• b = 4

• c = -1

Sum of zeroes

$\rm \implies \alpha + \beta = \dfrac{ - b}{a}$

$\rm \implies \alpha + \beta = \dfrac{ - 4}{4}$

$\rm \implies \alpha + \beta = - 1$

Product of zeroes

$\rm \implies \alpha \beta = \dfrac{ c}{a}$

$\rm \implies \alpha \beta = \dfrac{ -1}{4}$

We have:-

• α + β = -1

• αβ = -¼

Now we need to find the value of 1/α +1/β

$\rm \implies \dfrac{1}{ \alpha } + \dfrac{1}{ \beta }$

$\rm \implies \dfrac{ \beta + \alpha }{ \alpha \beta }$

$\rm \implies \dfrac{ \alpha+\beta }{ \alpha \beta }$

$\rm \implies \dfrac{ - 1 }{ \frac{ - 1}{4} }$

$\rm \implies - 1 \times \dfrac{4}{ - 1}$

$\rm \implies - 1 \times - 4$

$\rm \implies 4$

$\large\underline{ \boxed{ \rm \therefore \frac{1}{ \alpha } + \frac{1}{ \beta } = 4}}$