F.4 QUADRATIC EQUATION

1. Let p and q be constants. It is given that a and b are the roots of the equation 3x² - 5x + p = 0. If 1/a + 1/b are the roots of the equation 7x² + qx - 3 = 0, find the values of p and q.

a and b are the roots of 3x² - 5x + p = 0:
a + b = 5/3 …… (1)
ab = p/3 …… (2)

1/a + 1/b are the roots of 7x² + qx - 3 = 0
1/a + 1/b = -q/7 …… (3)
(1/a)(1/b) = -3/7 …… (4)

(4):
1/ab = -3/7
ab = -7/3 …… (5)

(2) = (5):
p/3 = -7/3
P = -7

(3):
(a + b)/ab = -q/7 …… (6)

Put (1) and (5) into (6):
(5/3) /(-7/3) = -q/7
-5/7 = -q/7
q = 5

Hence, p = -7, q = 5


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2. Let a and b be the roots of the equation x²+(p+3)+4=0, where p is a constant. Express a quadratuc equation in x with roots 2a+3 and 2b+3 in terms of p

a and b be the roots of the equation x² + (p + 3) + 4 = 0
a + b = -(p + 3)
ab = 4

(2a + 3) + (2b + 3)
= 2(a + b) + 6
= 2[-(p + 3)] + 6
= -2p - 6 + 6
= -2p

(2a + 3)(2b + 3)
= 4ab + 6(a + b) + 9
= 4(4) + 6[-(p + 3)] + 9
= 16 - 6p - 18 + 9
= 7 - p

Hence, the required equation is: x² + 2px + 7 - p = 0