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

Can reformulate as a cubic equation:

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

Square both sides to get:

Note that squaring can introduce spurious solutions, so we need to check later.

Subtract the right hand side from the left to get:

By the rational roots theorem, the only possible rational roots of this cubic are:

graph{4x^3-x^2+16x-64 [-0.698, 4.302, -1.18, 1.32]}

Then:

Then you can use Cardano's method to solve, finding a solution of the form:

hence

If you are really interested see:

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

- Start by distributing the 2 to both terms inside the parentheses: 2 * x * √x = 2x√x.
- Rewrite the equation as: 2x√x + x = 8.
- Subtract x from both sides of the equation: 2x√x = 8 - x.
- To isolate the term with the square root, divide both sides of the equation by 2: x√x = (8 - x) / 2.
- Square both sides of the equation to eliminate the square root: (x√x)^2 = [(8 - x) / 2]^2.
- Simplify the right side of the equation: x^3 = (8 - x)^2 / 4.
- Expand the right side of the equation: x^3 = (64 - 16x + x^2) / 4.
- Multiply both sides of the equation by 4 to eliminate the fraction: 4x^3 = 64 - 16x + x^2.
- Rearrange the equation to form a cubic equation: 4x^3 + x^2 - 16x - 64 = 0.
- 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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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