Question

.If α and β are zeros of the quadratic polynomial f(x) = x² -px +q, then the value of α²+β² *
2 points
2q - p²
p² - 2q
-p² - 2q
none of these​

Answer 1

Given, If α and β are zeros of the quadratic polynomial f(x) = x² -px +q

• To find, The value of α²+β²

Solution :

• As we know that

★ Sum of zeros = -(coefficient of x)/coefficient of x² = -b/a

★ Product of zeros = coefficient of constant term/coefficient of x² = c/a

• According to the given question

α and β are zeros of the quadratic polynomial

• Given polynomial : f(x) = x² -px +q

→ Sum of zeros = -b/a

→ α + β = -(-p)/1

→ α + β = p

Now,

→ Product of zeros = c/a

→ α × β = q/1

→ αβ = q

Value of α² + β²

→ α² + β²

→ (α + β)² - 2αβ

• Put the values

→ (p)² - 2 × q

→ p² - 2q

•°• Correct option is p² - 2q

Answer 2

Given :-

If α and β are zeros of the quadratic polynomial f(x) = x² -px +q

To Find :-

Value  of α²+β²

Solution :-

We know that

$\sf Sum \; of\; zeroes = \alpha +\beta = \dfrac{-b}{a}$

We have

b = -p

a = 1

$\sf \alpha +\beta =\dfrac{-(-p)}{1}$

$\sf \alpha +\beta = \dfrac{p}{1}$

$\bf \alpha +\beta=p$

$\sf Product \; of \;zeroes=\alpha \beta =\dfrac{c}a$

c = q

a = 1

$\sf \alpha \beta =\dfrac{q}{1}$

$\bf \alpha \beta =q$

Now

Finding the value

$\sf \alpha ^2+\beta ^2$

$\sf (\alpha +\beta )^2$

Apply identity

$\sf \alpha ^2-2\alpha \beta +\beta ^2$

$\sf (\alpha +\beta )^2+2\alpha \beta$

$\sf(p)^2-2\times q$

$\sf p^2-2\times q$

$\sf p^2-2q$