Question

if the sum of the zeroes of the polynomials p(x)=px2square-4x+2p is same as their products find the value of p
​

Answer 1

Given

$\sf\ px^2-4x+2p\\ \\ \sf\ \alpha+\beta= \alpha\beta$

To Find :-

we have to find the value of p

Solution :-

as we know

$\bullet\sf\ \alpha+\beta= \dfrac{-b}{a}\\ \\ \bullet\sf\ \alpha\beta= \dfrac{c}{a}\\ \\ \sf\ We\ have\ ,\\ \\ \sf\ \ on\ comparing\ the\ given\ equation\ with\ ax^2+bx+c\ \\ \\ \sf\ we\ get\ \ a= p\ \ ;\ b= -4\ \ ;\ c= 2p\\ \\ \\\star\sf\ \dfrac{-b}{a}= \dfrac{-(-4)}{p}\\ \\ \\\implies\sf\ \dfrac{-b}{a}= \dfrac{4}{p}-----(i)\\ \\ \\\star\sf\ \ \dfrac{c}{a}= \dfrac{2p}{p}\\ \\ \\\implies\sf\dfrac{c}{a}= 2 ------(ii)\\ \\ \\\sf\ as \ given \dfrac{-b}{a}=\dfrac{c}{a}\\ \\ \\ \sf\ Now\ from\ (i)\ and\ (ii)\\ \\ \\ :\implies\sf\ \dfrac{4}{p}= 2\\ \\ \\ :\implies\sf\ p= \cancel{\dfrac{4}{2}}\\ \\ \\ :\implies\underline{\boxed{\sf\ \ \ p= 2}}$

Answer 2

Answer:

Given :-

• The sum of the zeros of the polynomials is p(x) = px² - 4x + 2p is same as their products.

To Find :-

• What is the value of p.

Formula Used :-

➦ To find sum of zeros we know that,

${\red{\boxed{\small{\bold{\alpha + \beta =\: \dfrac{- b}{a}}}}}}$

➦ To find product of zeros we know that,

${\red{\boxed{\small{\bold{\alpha\beta =\: \dfrac{c}{a}}}}}}$

Solution :-

Given equation :

$\mapsto$ $\sf p{x}^{2} - 4x + 2p$

where, a = p , b = - 4 , c = 2p

First, we have to find the sum of zeros,

Given :

• - b = - 4
• a = p

According to the formula we get,

↦ $\sf \alpha + \beta =\: \dfrac{- (- 4)}{p}$

As we know that, [ ( - ) × ( - ) = ( + ) ]

➤ $\sf\bold{\pink{\alpha + \beta =\: \dfrac{4}{p}}}$

Again, we have to find the product of zeros,

Given :

• c = 2p
• a = p

According to the formula we get,

↦ $\sf \alpha\beta =\: \dfrac{2\cancel{p}}{\cancel{p}}$

➤ $\sf\bold{\pink{\alpha\beta =\: 2}}$

Now, according to the question,

⇒ $\sf \alpha + \beta =\: \alpha\beta$

We have,

• α + β = $\sf \dfrac{4}{p}$
• αβ = 2

⇒ $\sf \dfrac{4}{p} =\: 2$

By doing cross multiplication we get,

⇒ $\sf 2p =\: 4$

⇒ $\sf p =\: \dfrac{\cancel{4}}{\cancel{2}}$

➠ $\sf\bold{\purple{p =\: 2}}$

$\therefore$ The value of p is 2 .