SOLUTION: Solve 15x^4-31x^2+10=0 The solution of x= ??

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Solve 15x^4-31x^2+10=0 The solution of x= ??      Log On


   

Question 156148: Solve
15x^4-31x^2+10=0
The solution of x= ??
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let z=x%5E2. So z%5E2=%28x%5E2%29%5E2=x%5E4. In short, z%5E2=x%5E4

15x%5E4-31x%5E2%2B10=0 Start with the given equation


15z%5E2-31z%2B10=0 Plug in z=x%5E2 and z%5E2=x%5E4



Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for z:


Starting with the general quadratic


az%5E2%2Bbz%2Bc=0


the general solution using the quadratic equation is:


z+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 15%2Az%5E2-31%2Az%2B10=0 ( notice a=15, b=-31, and c=10)





z+=+%28--31+%2B-+sqrt%28+%28-31%29%5E2-4%2A15%2A10+%29%29%2F%282%2A15%29 Plug in a=15, b=-31, and c=10




z+=+%2831+%2B-+sqrt%28+%28-31%29%5E2-4%2A15%2A10+%29%29%2F%282%2A15%29 Negate -31 to get 31




z+=+%2831+%2B-+sqrt%28+961-4%2A15%2A10+%29%29%2F%282%2A15%29 Square -31 to get 961 (note: remember when you square -31, you must square the negative as well. This is because %28-31%29%5E2=-31%2A-31=961.)




z+=+%2831+%2B-+sqrt%28+961%2B-600+%29%29%2F%282%2A15%29 Multiply -4%2A10%2A15 to get -600




z+=+%2831+%2B-+sqrt%28+361+%29%29%2F%282%2A15%29 Combine like terms in the radicand (everything under the square root)




z+=+%2831+%2B-+19%29%2F%282%2A15%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




z+=+%2831+%2B-+19%29%2F30 Multiply 2 and 15 to get 30


So now the expression breaks down into two parts


z+=+%2831+%2B+19%29%2F30 or z+=+%2831+-+19%29%2F30


Lets look at the first part:


x=%2831+%2B+19%29%2F30


z=50%2F30 Add the terms in the numerator

z=5%2F3 Divide


So one answer is

z=5%2F3




Now lets look at the second part:


x=%2831+-+19%29%2F30


z=12%2F30 Subtract the terms in the numerator

z=2%2F5 Divide


So another answer is

z=2%2F5


So our solutions are:

z=5%2F3 or z=2%2F5






Remember, we let z=x%5E2. So this means that


5%2F3=x%5E2 or 2%2F5=x%5E2


Take the square root of both sides for each case (remember the "plus/minus")


x=sqrt%285%2F3%29, x=-sqrt%285%2F3%29, x=sqrt%282%2F5%29, or x=-sqrt%282%2F5%29



Rationalize the denominator (if necessary)

x=sqrt%2815%29%2F3, x=-sqrt%2815%29%2F3, x=sqrt%2810%29%2F5, or x=-sqrt%2810%29%2F5


--------------------------------------
Answer:

So the solutions are

x=sqrt%2815%29%2F3, x=-sqrt%2815%29%2F3, x=sqrt%2810%29%2F5, or x=-sqrt%2810%29%2F5