Math, asked by salonisarda12345, 1 year ago

if both x-2 and x-1/2 are factor of px2+5x + r show that p=r

Answers

Answered by aayushg1713

If x-2 and x-1/2 are factors of the equation this means that 2 and 1/2 are roots of the equation

In a quad equation product of roots = r/p = 2 × 1/2 = 1

=> p = r

aayushg1713: So if these 2 satisfy the equation this means that they are the roots of this equation.

aayushg1713: I will better explain you by an example

salonisarda12345: OK

aayushg1713: x^2 - 5x + 6 = 0 , this is a simple quardratic equation which has roots 2 and 3(you can calculate them)

aayushg1713: so now this means that you can write the quadratic equation as (x-2)(x-3) , you can expand this to verify

salonisarda12345: thanks

salonisarda12345: no I understand

aayushg1713: If this is our quadratic equation it means (x-2) and (x-3) are factors of it

salonisarda12345: now **

aayushg1713: :-)

Answered by abhi569

As both are the factors of the given polynomials, both must be zero for x = 2 and 1/2. Using factor theorem:

If x - 2 is factor: f(2) = 0

⇒ p(2)² + 5(2) + r = 0

⇒ 4p + 10 + r = 0 ...(1)

If x - 1/2 is factor: f(1/2) = 0

⇒ p(1/2)² + 5(1/2) + r = 0

⇒ p/4 + 5/2 + r = 0

⇒ p + 10 + 4r = 0 ...(2)

Subtract (1) from (2), we get:

⇒ 3p - 3r = 0

⇒ 3p = 3r

⇒ p = r proved

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