Question

Ifa and B are the zeros of the quadratic polynomial f(x) = x^2 -px +q, then find the
a2 +B2
​

Answer 1

Correct question

⇒If α and β are the zeroes of the quadratic polynomial f(x) = x² - px + q , then find α² + β²

Solution

we have

⇒x² - px + q = 0

We know that

⇒α + β = sum of the zeroes = -b/a

⇒αβ = product of the zeroes = c/a

By comparing with ax² + bx + c = 0

we get

⇒a = 1 , b = -p and c = q

Put the value on formula

⇒α + β = -(-p)/1 = p

⇒αβ = q/1 = q

We have to find the value of

⇒α² + β²

By simplify the equation , we get

⇒(α + β)² - 2αβ

Put the value

⇒(p)² - 2q

⇒p² - 2q

Answer

⇒p² - 2q

Answer 2

Answer:

Appropriate Question :-

• If α and β are the zeroes of the quadratic polynomial f(x) = x² - px + q, then find the value of α² + β².

Given :-

• α and β are the zeroes of the quadratic polynomial f(x) = x² - px + q.

To Find :-

• What is the value of α² + β².

Solution :-

Given equation :

$\dashrightarrow \sf\bold{x^2 - px + q}$

where,

• a = 1
• b = - p
• c = q

Now, we have to find the sum and product of the zeroes :

$\clubsuit$ Sum of Zeroes :

$\longmapsto \sf\boxed{\bold{\pink{Sum\: of\: Zeroes\: (\alpha + \beta) =\: \dfrac{- b}{a}}}}$

Then,

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

$\implies \sf \alpha + \beta =\: \dfrac{p}{1}\: \bigg\lgroup \bold{\purple{- \times - =\: +}} \bigg\rgroup$

$\implies \sf\bold{\green{\alpha + \beta =\: p}}$

Again,

$\clubsuit$ Product of Zeroes :

$\longmapsto \sf\boxed{\bold{\pink{Product\: of\: Zeroes\: (\alpha\beta) =\: \dfrac{c}{a}}}}$

Then,

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

$\implies \sf\bold{\green{\alpha\beta =\: q}}$

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

As we know that :

$\longmapsto \sf\boxed{\bold{\pink{a^2 + b^2 =\: (a + b)^2 - 2ab}}}$

We have :

• α + β = p
• αβ = q

According to the question by using the formula we get,

$\leadsto \sf \alpha^2 + \beta^2 =\: (\alpha + \beta)^2 - 2\alpha\beta$

$\leadsto \sf \alpha^2 + \beta^2 =\: (p)^2 - 2q$

$\leadsto \sf\bold{\red{\alpha^2 + \beta^2 =\: p^2 - 2q}}$

$\therefore$ The value of α² + β² is p² - 2q.