How do you solve #2(xsqrt x) + x=8#?

Question

Abigail Allen · Accepted Answer

Can reformulate as a cubic equation:

#4x^3-x^2+16x-64 = 0#

which has an irrational root findable using Cardano's method or similar.

Subtract #x# from both sides of the equation to get: #2xsqrt(x) = 8-x# Square both sides to get: #4x^3 = (8-x)^2 = 64 - 16x+x^2# Note that squaring can introduce spurious solutions, so we need to check later. Subtract the right hand side from the left to get: #f(x) = 4x^3-x^2+16x-64 = 0# By the rational roots theorem, the only possible rational roots of this cubic are: #+-64#, #+-32#, #+-16#, #+-8#, #+-4#, #+-2#, #+-1#, #+-1/2#, #+-1/4# None of these are roots (though #2# comes quite close). graph{4x^3-x^2+16x-64 [-0.698, 4.302, -1.18, 1.32]} It is possible to solve #f(x) = 0# algebraically, but probably beyond the scope of your course. First let #x_1 = x - 1/12# Then: #4x_1^3 = 4(x-1/12)^3 = 4(x^3-x^2/4+x/48-1/1728)# #=4x^3-x^2+x/16-1/432# So #f(x) = 4x^3-x^2+16x-64# #= 4x_1^3 + (16-1/16)x+(1/432-64)# #= 4x_1^3 + 255/16x_1+(1/432-64+255/(16*12))# This is of the form #ax_1^3+bx_1+c# Then you can use Cardano's method to solve, finding a solution of the form: #x_1 = root(3)(A+sqrt(B)) + root(3)(A-sqrt(B))# hence #x = root(3)(A+sqrt(B)) + root(3)(A-sqrt(B)) + 1/12# If you are really interested see: https://tutor.hix.ai

Isaiah Anthony · Answer

undefined To solve the equation 2(x√x) + x = 8, we can follow these steps:

1. Start by distributing the 2 to both terms inside the parentheses: 2 * x * √x = 2x√x.
2. Rewrite the equation as: 2x√x + x = 8.
3. Subtract x from both sides of the equation: 2x√x = 8 - x.
4. To isolate the term with the square root, divide both sides of the equation by 2: x√x = (8 - x) / 2.
5. Square both sides of the equation to eliminate the square root: (x√x)^2 = [(8 - x) / 2]^2.
6. Simplify the right side of the equation: x^3 = (8 - x)^2 / 4.
7. Expand the right side of the equation: x^3 = (64 - 16x + x^2) / 4.
8. Multiply both sides of the equation by 4 to eliminate the fraction: 4x^3 = 64 - 16x + x^2.
9. Rearrange the equation to form a cubic equation: 4x^3 + x^2 - 16x - 64 = 0.
10. Solve the cubic equation using factoring, synthetic division, or numerical methods to find the values of x.

Note: The solution to the cubic equation may involve complex numbers or irrational roots, depending on the specific values of x.