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Find the Value of P for Which the Quadratic Equation (2p + 1)X^2 − (7p + 2)X + (7p − 3) = 0 Has Equal Roots. Also Find These Roots. - Mathematics

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Find the value of p for which the quadratic equation (2p + 1)x2 − (7p + 2)x + (7p − 3) = 0 has equal roots. Also find these roots.

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Solution

Given quadratic equation:
(2p+1)x2(7p+2)x+(7p3)=0

To have equal roots, the discriminant should be zero.

∴ D = 0
⇒ (7p+2)2−4×(2p+1)×(7p−3)=0
⇒49p2+28p+4−4×(14p2−6p+7p−3)=0
⇒49p2+28p+4−56p2−4p+12=0
⇒−7p2+24p+16=0

⇒7p2−24p−16=0
⇒7p2−28p+4p−16=0

⇒7p(p−4)+4(p−4)=0

⇒(p−4)(7p+4)=0
⇒p=4 or −4/7
Therefore, the values of p for which the given equation has equal roots are 4, −4/7.

For p = 4:
(2p+1)x2−(7p+2)x+(7p−3)=0
⇒9x2−30x+25=0⇒(3x−5)2=0
⇒x=5/3, 5/3

For p = −4/7:
(2p+1)x2−(7p+2)x+(7p−3)=0
⇒(2(−47)+1)x2−(7(−47)+2)x+(7(−47)−3)=0
⇒(−17)x2+2x−7=0⇒17x2−2x+7=0
⇒x2−14x+49=0⇒(x−7)2=0
⇒x=7, 7

Thus, the equal roots are 7 or 5/3.

Concept: Nature of Roots
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