tangaloomaflyer
New member
- Joined
- Apr 24, 2016
- Messages
- 13
Hi,
I'm not sure if this is a problem with my arithmetic or if I misunderstand the question. I'll write out the whole question with the answers to the first two parts, but it is the third part I am stuck on:
Given that p and q are roots of the equation 3x2 +x + 2 = 0
(i) evaluate 1/p2 + 1/q2
Answer: -11/4
working:
1/p2 +1/q2 = (p+q)2 - 2pq / (pq)2
Re-arrange 3x2 +x + 2 = 0 tothe form x2 - (p+q)x + (pq) = 0
p+q = 1/3
pq = 2/3
[(p+q)2 - 2pq] / (pq)2 = (1/9 - 4/3) / (4/9) = -11/4
(ii) Form an equation with roots 1/p2 and 1/q2
Answer: 4x2 +11x + 9 = 0Working:
Make new roots P and QUsing the form x2 - (P+Q)x + (PQ) = 0 equivalent to x2 - (b/a)x + (c/a) = 0
P+Q = 1/p2 + 1/q2 =-11/4 =-b/a therefore b=11, a=4
(pq)2 = 4/9 from [(p+q)2 - 2pq] / (pq)2 = (1/9 - 4/3) / (4/9)PQ = 1/(pq)2 = 9/4 = c/a
therefore a=4, b=11, c=9
equation is 4x2 + 11x + 9 = 0
(iii) Show that 27p4 = 11p + 10
Answer: ???
I am assuming that p is the root of the original equation 3x2 +x + 2 = 0
From the discriminant of the quadratic formula, sqrt(b2 - 4ac) = sqrt(-23), therefore the roots are not real. When I find the values of 27p4 and 11p + 10 for the two possible roots I get different values. I think this is due to my poor fractional arithmetic of complex numbers because I keep getting different values! I am sure there must be a simpler way.
My workings (apologies as I don't know how to write the radical sign or fractions using code):
Find roots using quadratic formula:
p = [-1 + sqrt(23)i]/6 or [-1 - 1 sqrt(23)i]/6
Find 27p4
p2 = (1 + 2sqrt(23)i +23)/36 or (1 - 2sqrt(23)i +23)/36
p4 = (2sqrt(23)i +24)2 / 362 or (-2sqrt(23)i +24)2 / 362
= (96sqrt(23)i + 484) / 1296
27p4 = 2sqrt(23)i + 242/24
find 11p +10
11p = -11/6 + 11/6 sqrt(23)i or -11/6 - 11/6 sqrt(23)i
11p+ 10 does not equal 2sqrt(23)i + 242/24
Any ideas? The question is from a section of the textbook on quadratics, so I'm thinking I need to make the equation (p2)(27p2)= 11p +10 and use the answers of the first two parts to logically prove the equation, but I'm stuck. Unless the p is referring to a root of 4x2 + 11x + 9 = 0 ?
I'm not sure if this is a problem with my arithmetic or if I misunderstand the question. I'll write out the whole question with the answers to the first two parts, but it is the third part I am stuck on:
Given that p and q are roots of the equation 3x2 +x + 2 = 0
(i) evaluate 1/p2 + 1/q2
Answer: -11/4
working:
1/p2 +1/q2 = (p+q)2 - 2pq / (pq)2
Re-arrange 3x2 +x + 2 = 0 tothe form x2 - (p+q)x + (pq) = 0
p+q = 1/3
pq = 2/3
[(p+q)2 - 2pq] / (pq)2 = (1/9 - 4/3) / (4/9) = -11/4
(ii) Form an equation with roots 1/p2 and 1/q2
Answer: 4x2 +11x + 9 = 0Working:
Make new roots P and QUsing the form x2 - (P+Q)x + (PQ) = 0 equivalent to x2 - (b/a)x + (c/a) = 0
P+Q = 1/p2 + 1/q2 =-11/4 =-b/a therefore b=11, a=4
(pq)2 = 4/9 from [(p+q)2 - 2pq] / (pq)2 = (1/9 - 4/3) / (4/9)PQ = 1/(pq)2 = 9/4 = c/a
therefore a=4, b=11, c=9
equation is 4x2 + 11x + 9 = 0
(iii) Show that 27p4 = 11p + 10
Answer: ???
I am assuming that p is the root of the original equation 3x2 +x + 2 = 0
From the discriminant of the quadratic formula, sqrt(b2 - 4ac) = sqrt(-23), therefore the roots are not real. When I find the values of 27p4 and 11p + 10 for the two possible roots I get different values. I think this is due to my poor fractional arithmetic of complex numbers because I keep getting different values! I am sure there must be a simpler way.
My workings (apologies as I don't know how to write the radical sign or fractions using code):
Find roots using quadratic formula:
p = [-1 + sqrt(23)i]/6 or [-1 - 1 sqrt(23)i]/6
Find 27p4
p2 = (1 + 2sqrt(23)i +23)/36 or (1 - 2sqrt(23)i +23)/36
p4 = (2sqrt(23)i +24)2 / 362 or (-2sqrt(23)i +24)2 / 362
= (96sqrt(23)i + 484) / 1296
27p4 = 2sqrt(23)i + 242/24
find 11p +10
11p = -11/6 + 11/6 sqrt(23)i or -11/6 - 11/6 sqrt(23)i
11p+ 10 does not equal 2sqrt(23)i + 242/24
Any ideas? The question is from a section of the textbook on quadratics, so I'm thinking I need to make the equation (p2)(27p2)= 11p +10 and use the answers of the first two parts to logically prove the equation, but I'm stuck. Unless the p is referring to a root of 4x2 + 11x + 9 = 0 ?