Solve $x^4+(4-x)^4=32$ Equation

  • MHB
  • Thread starter solakis1
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In summary, the conversation discusses an equation x^4+(4-x)^4=32 and how to solve it. The equation can be solved by expanding the fourth powers and simplifying it, resulting in a quadratic equation. The steps to solve the equation include moving all terms to one side, factoring or using the quadratic formula, and checking the solutions. There are two special cases where the solutions will be imaginary numbers. The equation can also be solved by factoring, but this method may not always be possible.
  • #1
solakis1
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0
solve the following equation:

$x^4+(4-x)^4=32$
 
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  • #2
My attempt.
Looks like symmetry would be helpful.
Let's set $x=y+2$.
Then we get
$$(y+2)^4 + (4-(y+2))^4=(y+2)^4 + (y-2)^4=2y^4+48y^2+32=32 \implies y^2(y^2+24)=0\implies y=0 \implies x=2$$
 
  • #3
solakis said:
solve the following equation:

$x^4+(4-x)^4=32$

Do an average substitution $t=x-2$. I have a solution to a similar equation:

 

What is the equation for $x^4+(4-x)^4=32$?

The equation is a quartic equation, which means it has a degree of 4 and can be written as $x^4-4x^3+6x^2-4x+16=0$.

What is the degree of this equation?

The degree of this equation is 4, since the highest power of the variable $x$ is 4.

How many solutions does this equation have?

This equation has 4 complex solutions, which means it has 4 values of $x$ that satisfy the equation. However, not all of these solutions may be real numbers.

What is the process for solving this equation?

The first step is to simplify the equation by expanding the fourth powers. Then, you can rearrange the equation to isolate the terms with $x$ on one side. Next, you can use the quadratic formula to solve for the values of $x$. Finally, you can substitute these values back into the original equation to check for solutions.

Can this equation be solved without using the quadratic formula?

Yes, it is possible to solve this equation without using the quadratic formula. However, the process may be more complicated and time-consuming. You can also use graphing or numerical methods to approximate the solutions.

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