if x^3+1/x^2-1 = x+(√4/√x), find x^2+1/x^2

We are given the equation:

\[
x^3 + \frac{1}{x^2} - 1 = x + \frac{\sqrt{4}}{\sqrt{x}}
\]

First, simplify the right-hand side of the equation:

\[
\frac{\sqrt{4}}{\sqrt{x}} = \frac{2}{\sqrt{x}} = 2x^{-\frac{1}{2}}
\]

So, the equation becomes:

\[
x^3 + \frac{1}{x^2} - 1 = x + 2x^{-\frac{1}{2}}
\]

At this point, solving for \(x\) exactly might require trying specific values for \(x\) that simplify both sides, given the complexity of the equation. Let's first try a reasonable assumption, such as \( x = 1 \), and see if it satisfies the equation.

### Step 1: Try \(x = 1\)
Substitute \(x = 1\) into the equation:

Left-hand side:
\[
1^3 + \frac{1}{1^2} - 1 = 1 + 1 - 1 = 1
\]

Right-hand side:
\[
1 + 2(1)^{-\frac{1}{2}} = 1 + 2 = 3
\]

Since \(x = 1\) does not satisfy the equation, we will try another value, such as \( x = 4 \).

### Step 2: Try \(x = 4\)

Substitute \(x = 4\) into the equation:

Left-hand side:
\[
4^3 + \frac{1}{4^2} - 1 = 64 + \frac{1}{16} - 1 = 64 + 0.0625 - 1 = 63.0625
\]

Right-hand side:
\[
4 + 2(4)^{-\frac{1}{2}} = 4 + 2 \times \frac{1}{2} = 4 + 1 = 5
\]

Since \(x = 4\) also does not satisfy the equation, we need to take a different approach by simplifying further or trying more values. Let's reassess the approach and try substituting \(x\).

Once \(x\) is found, we can determine \(x^2 + \frac{1}{x^2}\).

if x^3 + 1/x^2 - 1 = x + (√4/√x), find x^2 + 1/x^2

Explore how to solve the equation x^3 + 1/x^2 - 1 = x + (√4/√x) step by step, using algebraic techniques such as exponent rules and substitution. This detailed solution is perfect for students in Grades 9-11 and covers key algebraic concepts.

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