Question
Download Solution PDFIf both the roots of the quadratic equation (2p + 1) x2 + (3p + 2)x + (p+1) = 0 and (6p + 1)x2 + 9px + 3p - 1 = 0 are common, then what is the value of p2 +3p-1?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
Quadratic equation 1: (2p + 1) x2 + (3p + 2)x + (p+1) = 0
Quadratic equation 2: (6p + 1)x2 + 9px + 3p - 1 = 0
Both quadratic equations have common roots.
Formula Used:
If two quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have both roots common, then
Calculations:
Comparing the coefficients of the given quadratic equations:
a1 = 2p + 1, b1 = 3p + 2, c1 = p + 1
a2 = 6p + 1, b2 = 9p, c2 = 3p - 1
Since both roots are common, we have:
First, equate the first two ratios:
⇒ 9p(2p + 1) = (3p + 2)(6p + 1)
⇒ 18p2 + 9p = 18p2 + 3p + 12p + 2
⇒ 18p2 + 9p = 18p2 + 15p + 2
⇒ 9p - 15p = 2
⇒ -6p = 2
⇒ p = -2/6 = -1/3
Now, let's verify if this value of p satisfies the equality of the second and third ratios:
Substitute p = -1/3:
The value of p = -1/3 satisfies the condition for common roots.
Now, we need to find the value of p2 + 3p - 1.
Substitute p = -1/3:
p2 + 3p - 1 =
⇒ p2 + 3p - 1 =
⇒ p2 + 3p - 1 =
⇒ p2 + 3p - 1 =
⇒ p2 + 3p - 1 =
∴ The value of p2 + 3p - 1 is -17/9.
Last updated on Feb 8, 2025
-> OSSC Excise SI PET/PMT Merit List has been released on the official website. The test was conducted on 4th and 5th February 2025.
-> OSSC Excise SI 2024 Notification has been released for 10 vacancies.
-> The selection process includes Written Examination, PMT and PET, and Certificate Verification.