The equation x^2 + 4x + p = 0 where p is a real constant, has distinct real roots M and N.
If M^2 + N^2 +M^2N^2 + 3(M+N)-19=0 find value of p.
A math
2008-10-29 5:48 am
回答 (3)
2008-11-02 6:01 am
sum of roots = M+N = -4
product of roots = MN = p
M^2 + N^2 +M^2N^2 + 3(M+N)-19=0
(M^2 + 2MN + N^2) - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(M+N)^2 - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(-4)^2 - 2p + p^2 + 3*(-4) - 19 = 0
p^2 - 2p - 15 = 0
p = (2-(2^2 - 4*(1)*(-15))^0.5)/2 or (2+(2^2 - 4*(1)*(-15))^0.5)/2
p = -3 or 5 (rej.)
Note that p = 5 is rejected because the equation x^2 + 4x + 5 = 0 has discriminant = 4^2 - 4*1*5 = -4 < 0, having no real roots at
product of roots = MN = p
M^2 + N^2 +M^2N^2 + 3(M+N)-19=0
(M^2 + 2MN + N^2) - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(M+N)^2 - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(-4)^2 - 2p + p^2 + 3*(-4) - 19 = 0
p^2 - 2p - 15 = 0
p = (2-(2^2 - 4*(1)*(-15))^0.5)/2 or (2+(2^2 - 4*(1)*(-15))^0.5)/2
p = -3 or 5 (rej.)
Note that p = 5 is rejected because the equation x^2 + 4x + 5 = 0 has discriminant = 4^2 - 4*1*5 = -4 < 0, having no real roots at
2008-10-29 5:58 am
sum of roots = M+N = -4
product of roots = MN = p
M^2 + N^2 +M^2N^2 + 3(M+N)-19=0
(M^2 + 2MN + N^2) - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(M+N)^2 - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(-4)^2 - 2p + p^2 + 3*(-4) - 19 = 0
p^2 - 2p - 15 = 0
p = (2-(2^2 - 4*(1)*(-15))^0.5)/2 or (2+(2^2 - 4*(1)*(-15))^0.5)/2
p = -3 or 5 (rej.)
Note that p = 5 is rejected because the equation x^2 + 4x + 5 = 0 has discriminant = 4^2 - 4*1*5 = -4 < 0, having no real roots at all.
product of roots = MN = p
M^2 + N^2 +M^2N^2 + 3(M+N)-19=0
(M^2 + 2MN + N^2) - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(M+N)^2 - 2MN + (MN)^2 + 3(M+N) - 19 = 0
(-4)^2 - 2p + p^2 + 3*(-4) - 19 = 0
p^2 - 2p - 15 = 0
p = (2-(2^2 - 4*(1)*(-15))^0.5)/2 or (2+(2^2 - 4*(1)*(-15))^0.5)/2
p = -3 or 5 (rej.)
Note that p = 5 is rejected because the equation x^2 + 4x + 5 = 0 has discriminant = 4^2 - 4*1*5 = -4 < 0, having no real roots at all.
2008-10-29 5:57 am
Sum of roots:M+N=-4
Product of roots:MN=p
M2 + N2 +M2 N2 + 3(M+N)-19=0
[(M+N)2 -2MN]+(MN)2 +3(M+N)-19=0
[(-4)2 -2p]+p2 +3(-4)-19=0
(16-2p)+p2 -12-19=0
p2 -2p-15=0
p=5 or p=-3
Product of roots:MN=p
M2 + N2 +M2 N2 + 3(M+N)-19=0
[(M+N)2 -2MN]+(MN)2 +3(M+N)-19=0
[(-4)2 -2p]+p2 +3(-4)-19=0
(16-2p)+p2 -12-19=0
p2 -2p-15=0
p=5 or p=-3
收錄日期: 2021-04-23 20:37:33
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