Question

Solve for \[c^2 + 5d^2\]: \[a^2 + 5b^2 = 10 \\ bc - ad = 5 \\ ac + 5bd = 15\]

Answer 1

from second equation \( 5(bc-ad)^2=125 \) and from third u have \( (ac+5bd)^2=225 \)

expand and then add this 2 equation and tell me what u get?

Answer 2

\[5(bc)^2 + 5(ad^2) - 10abcd = 125 \\ (ac)^2 + 25(bd)^2 + 10abcd= 225\]

Is this correct so far?

Answer 3

Quite right ; add them

Answer 4

\[0(ac)^2 + 5(bc)^2 + 5(ad^2) + 0(bd)^2 - 10abcd = 125 \\ (ac^2) + 0(bc)^2 + 0(ad)^2 + 25(bd)^2 + 10abcd = 225\]\[(ac^2) + 5(bc)^2 + 5(ad^2) + 25(bd)^2 = 350\]

Is this right so far?

In the beginning, how did you know to use these two equations and also to square and multiply them by a constant?

Answer 5

well thats experience of problem solving; and u will get this skill through involvement such problems ; i did that because of form of a^2+5b^2=10

Answer 6

c^2(.........

Answer 7

I managed to expand it to\[a^2c^2 + 5b^2c^2 +5a^2d^2 +25b^2d^2 = 350\]I am rusty with my factoring...how do I take out both \[a^2+5b^2\] at the same time?

Answer 8

Oh wait, let me try something else

Answer 9

\[a^2(c^2 + 5d^2) + 5b^2(c^2 + 25d^2)\]\[(a^2 + 5b^2)(c^2 + 5d^2)(c^2 + 25d^2)\]
Am I allowed to do this?

Answer 10

be carefull thats \(a^2(c^2+5d^2)+5b^2(c^2+5d^2) \)

Answer 11

Oh right, I forgot to factor out the '5' from '25'

Answer 12

I see now...\[(a^2 + 5b^2)(c^2 + 5d^2)^2\]I can do this?

Answer 13

I don't think the square belongs there, actually

Answer 14

So \[(10)(c^2 + 5d^2) = 350\]and I need to find \[(c^2 + 5d^2)\] so divide, right?

Answer 15

omg mukushla you are a genius...I wish I could know how to solve this without help (the squaring and multiplying in the beginning)

Answer 16

yes

Answer 17

Answer: 35

Answer 18

Quite right

Answer 19

u will get there just keep on trying and solving many many problems like this ;)