Question

if alpha and beta are the zeroes of the polynomial x2+x-2,find the value of alpha square+beta square

Answer 1

Answer:

5

Step-by-step explanation:

Given a quadratic equation such that,

${x}^{2} + x - 2 = 0$

Also, its given that,

The zeroes of this Polynomial are,

• $\alpha$
• $\beta$

Now, we know that,

Sum of zeroes = -b/a

$= > \alpha + \beta = - \frac{1}{1} \\ \\ = > \alpha + \beta = - 1$

And, we know that,

Product of zeroes = c/a

$= > \alpha \beta = \frac{ - 2}{1} \\ \\ = > \alpha \beta = - 2$

Now, to find the value of ,

${ \alpha }^{2} + { \beta }^{2}$

Therefore, we will get,

$= {( \alpha + \beta )}^{2} - 2 \alpha \beta$

Substituting the values, we get,

$= {( - 1)}^{2} - 2( - 2) \\ \\ = 1 + 4 \\ \\ = 5$

Hence, required value is 5.

Answer 2

Given: α and β are the zeros of the polynomial x² + x - 2.

To find: The value of α² + β².

Answer:

We know that the general form of an equation is ax² + bx + c, where:

• The  sum of the zeros is given by -b/a.
• The product of the zeros is given by c/a.

From the given equation, we have:

• a = 1
• b = 1
• c = -2

This implies that:

Sum of the zeros: α + β = -b/a = -1/1 = -1

Product of the zeros: αβ = c/a = -2/1 = -2

Now, we've to find the value of α² + β².

We know that α² + β² = (α + β)² - 2αβ.

Therefore, substituting the values, we get:

α² + β² = (-1)² - 2(-2)

α² + β² = 1 + 4

α² + β² = 5

Therefore, the value of α² + β² is 5.