SOLUTION: Please help. Solve the following 15x^4-28x^2+5=0. I know that eventually I have to use the quadratic equation, but I am unsure of how to get the problem into a standard form to use

Question 177410: Please help. Solve the following 15x^4-28x^2+5=0. I know that eventually I have to use the quadratic equation, but I am unsure of how to get the problem into a standard form to use. Thank you.
Found 3 solutions by jim_thompson5910, stanbon, solver91311:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Let . So this means that

Replace with and with "z"

Notice we have a quadratic equation in the form of where , , and

Let's use the quadratic formula to solve for z

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

-----------------------------------------

Since we let , we need to plug that in:

Start with the first solution for "z"

Plug in

Take the square root of both sides.

or Break up the "plus/minus".

or Simplify the square root.

So the first two solutions are or

------------------------------------------------

Start with the second solution for "z"

Plug in

Take the square root of both sides.

or Break up the "plus/minus".

or Simplify the square root.

So the next two solutions are or

============================================================

Answer:

So the four solutions are:

, , or
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
15x^4-28x^2+5=0
---------
15x^4 - 25x^2 - 3x^2 + 5 = 0
5x^2(3x^2 - 5) - (3x^2 - 5) = 0
(3x^2 - 5)(5x^2 - 1) = 0
x^2 = 5/3 or x^2 = 1/5
x = +/-sqrt(5/3) or x = +/-sqrt(1/5)
========================================
Cheers,
Stan H.
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!

The trick is to let . That gives you which is a quadratic in standard form.

Look at the discriminant . Since the discriminant is a perfect square, the quadratic has rational roots and can be factored:

Hence: or

But remember so:

or

or

or

Which is 4 roots as you would expect with a quartic.
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