Solving x^4 - x^2 = k for x: Tips and Tricks to Simplify Equations

  • Thread starter Ted123
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In summary, to simplify the equation x^4 - x^2 = k, you can factor out the common term x^2 and use the quadratic formula to solve for x^2 and then take the square root to find the values of x. The first step in solving this equation is to rearrange it to get it in the form x^4 - x^2 - k = 0. Yes, the quadratic formula can be used to solve x^4 - x^2 = k, but you will need to take the square root of both solutions for x^2. Other methods for solving this equation include using the substitution method and checking your solution by plugging it back into the original equation or graphing it.
  • #1
Ted123
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Solve the equation [tex]x^4 - x^2 = k[/tex] where [itex]k>0[/itex].

Am I being thick or how do I solve this for [itex]x[/itex]?

Factorising gives [tex]x^2(x^2-1)=k[/tex] but now where?
 
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  • #2
Let [itex]y= x^2[/itex] and that becomes a quadratic function:
[itex]y^2- y= k[/itex] or [itex]y^2- y- k= 0[/itex]. Solve that for y, then solve [itex]x^2= y[/itex] for x.
 

1. How do I simplify the equation x^4 - x^2 = k?

To simplify this equation, you can rewrite it as x^2(x^2 - 1) = k and then factor out the common term x^2. This will leave you with x^2 = k / (x^2 - 1). You can then use the quadratic formula to solve for x^2 and then take the square root to find the values of x.

2. What is the first step in solving x^4 - x^2 = k?

The first step is to rearrange the equation to get it in the form x^4 - x^2 - k = 0. This makes it easier to factor and solve for x.

3. Can I use the quadratic formula to solve x^4 - x^2 = k?

Yes, you can use the quadratic formula to solve this equation. However, you will end up with two solutions for x^2, so you will need to take the square root of both solutions to get the values for x.

4. Are there any other methods for solving x^4 - x^2 = k?

Yes, you can also use the substitution method by letting y = x^2 and solving for y. Once you have the values of y, you can then substitute them back into the original equation to find the values of x.

5. How can I check my solution for x^4 - x^2 = k?

You can check your solution by plugging the values of x back into the original equation and seeing if it equals the given value of k. You can also graph the equation and see if the x-intercepts match your solutions.

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