the quartic equation x^4+8x^3=9x^2+20x+3
how to complete the square on the left hand side? I tried to factor out an x^2 first
http://mathworld.wolfram.com/QuarticEquation.html
To complete the square on the left-hand side of the equation x^4 + 8x^3 = 9x^2 + 20x + 3, we need to rearrange the terms and separate the perfect square terms from the remaining terms. Here's how you can go about it:
1. Start with the original equation: x^4 + 8x^3 = 9x^2 + 20x + 3.
2. Move all the terms to one side of the equation to have the quadratic and linear terms together: x^4 + 8x^3 - 9x^2 - 20x - 3 = 0.
3. Now, consider the quartic equation x^4 + 8x^3 - 9x^2 - 20x - 3 = 0. We want to focus on completing the square for the terms involving x^2 and x respectively.
4. For the x^2 terms (x^4 - 9x^2), we can complete the square by adding (b/2)^2 to both sides, where b is the coefficient of x^2. In this case, b = -9, so we add (-9/2)^2 = 81/4 to both sides. We obtain:
x^4 + 8x^3 - 9x^2 + 81/4 - 20x - 3 = 81/4.
5. For the x terms (-20x), we can complete the square by adding (b/2)^2 to both sides, where b is the coefficient of x. In this case, b = -20, so we add (-20/2)^2 = 100 to both sides. We obtain:
x^4 + 8x^3 - 9x^2 - 20x + 81/4 - 3 + 100 = 81/4 + 100.
6. Simplify the equation:
x^4 + 8x^3 - 9x^2 - 20x + 181/4 = 425/4.
7. Notice that the left-hand side can now be factored as a perfect square:
(x^2 + 4x - 9/2)^2 = 425/4.
Now, you can go ahead and solve for x by taking the square root of both sides of the equation and solving for x. However, solving quartic equations can be quite complex and often involve numerical methods or factoring techniques.