Two functions intersect,solve the equation.

Question

Given two functions,show where they intersect

$(x^2−5)^2/(x+7)^2=\sqrt{169-x^2}$

I have already tried to square both of them but I get a very complex equation and I can not solve it. I saw a guy who put Ln before and before the two sides of the equation.

You didn't provide two functions; you gave one equation. Do you mean to say the first function is $(x^2-5)^2/(x+7)^2$ and the second function is $\sqrt{169-x^2}$? — JRN, Apr 30 '14 at 02:41
yea, dude if you speak spanish let me know, its difficult to explain with English — Rocket, Apr 30 '14 at 02:42
Sorry, I don't speak Spanish. Wolfram Alpha says there are two intersections: $x\approx -3.96135$ and $x\approx 7.19026$. — JRN, Apr 30 '14 at 02:46
ok dude, but do you think that applying natural logarithm is ok? — Rocket, Apr 30 '14 at 02:48
I haven't tried it yet. Do you know the laws of logarithms? Use these to try to get the logarithms of both sides of the equations; see if you get a simpler expression. — JRN, Apr 30 '14 at 02:51
Your equation is a 8th-degree polynomial. There is no simple solution for such equations. See How to solve an nth degree polynomial equation. — Biswajit Banerjee, Apr 30 '14 at 03:00
dude and do you agree with taking Natural Logarithm?, should i move this question to another section ? — Rocket, Apr 30 '14 at 03:01
Natural logarithms will not help. I suggest you make sure that the equation you have written is correct. — Biswajit Banerjee, Apr 30 '14 at 03:02
Is this homework? If so, what topic is it about? Does your class allow the use of graphing calculators? If so, then perhaps that would be the way to answer this question. — JRN, Apr 30 '14 at 03:05
diferential calculus, but we are supposed to do it without graphs — Rocket, Apr 30 '14 at 03:07
Are you required to plot their graphs and find number of solutions? — evil999man, Apr 30 '14 at 03:07
Because at best you will get something like $$ 4\left[\ln(x-\sqrt{5}) + \ln(x+\sqrt{5})- \ln(x+7)\right] = \ln(13-x) + \ln(13+x)$$ which cannot be solved without numerical methods. — Biswajit Banerjee, Apr 30 '14 at 03:09

score 1 · Accepted Answer · answered Apr 30 '14 at 03:06

1

This equation simplifies to an 8th degree polynomial that can't be solved explicitly. Applying logarithms gives you $2 \log(x^2 - 5) - 2 \log(x+7) = \frac{1}{2} \log(169-x^2)$, but this does not simplify. As always, numerical approximations such as newton's method will get you approximate answers that @JoelReyesNoche gave.

answered Apr 30 '14 at 03:06

qwr

10,716

ok it seems like it is the answer , thank you @JoelReyesNoche, and you qwr, so in conclusion you take natural logarithm . Do you know a good book for this topic, i have Stewart´s Precalculus, but it doesn´t explain it well – Rocket Apr 30 '14 at 03:19
You're welcome. But the conclusion is not to take natural logarithms; the solution is to use numerical approximations. – JRN Apr 30 '14 at 03:51

Two functions intersect,solve the equation.

1 Answers1