Question

if alpha and beta are the roots of the equation x^+2x+8=0 then the value of alpha/beta + beta / alpha is

Answer 1

Answer:

The required value is "-1.5"

Step-by-step explanation:

Given :

α and β are the roots of the equation x² + 2x + 8 = 0

To find :

the value of  $\sf \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}$

Solution :

⇝ Relation between the sum of zeroes and coefficients :

• Sum of zeroes = -(x coefficient)/x² coefficient

• Product of zeroes = constant term/x² coefficient

For the given quadratic equation x² + 2x + 8 = 0,

• x² coefficient = 1
• x coefficient = 2
• constant term = 8

Therefore,

➺ α + β = -2

➺ αβ = 8

Let's simplify  $\dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}$

   $\sf \implies \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= \bigg(\dfrac{\alpha}{\beta}\times \dfrac{\alpha}{\alpha}\bigg)+\bigg(\dfrac{\beta}{\alpha} \times \dfrac{\beta}{\beta}\bigg) \\\\\\\sf \implies \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha} =\dfrac{\alpha^2}{\alpha\beta}+\dfrac{\beta^2}{\alpha \beta} \\\\\\ \sf \implies \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= \dfrac{\alpha^2+\beta^2}{\alpha \beta}$

we have the value of αβ

Now, we have to find the value of ( α² + β² )

We know,

$\boxed{\bf (a+b)^2=a^2+b^2+2ab}$

Put a = α and b = β,

(α + β)² = α² + β² + 2αβ

(-2)² = α² + β² + 2(8)

4 = α² + β² + 16

α² + β² = 4 - 16

α² + β² = -12

We have the required values...

Substitute them;

  $\sf \longrightarrow \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= \dfrac{\alpha^2+\beta^2}{\alpha \beta} \\\\ \sf \longrightarrow \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= \dfrac{-12}{8} \\\\ \sf \longrightarrow \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= \dfrac{-3 \times 4}{2 \times 4} \\\\ \sf \longrightarrow \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= \dfrac{-3}{2} \\\\ \sf \longrightarrow \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}= -1.5$

Therefore, the required answer is "-1.5"