To solve the equation \(x^2 + \sqrt{x} - 10 = 0\), let's proceed step by step.

### Step 1: Substitution
Let us make a substitution to simplify the equation. Set \( y = \sqrt{x} \), which means \( x = y^2 \). Substituting into the original equation:

\[
y^4 + y - 10 = 0
\]

### Step 2: Solve the quartic equation
We now need to solve \( y^4 + y - 10 = 0 \). This equation is a bit more complex as it involves a fourth power, and there are no simple factorization techniques that immediately apply here. The equation could be solved numerically or using approximation methods.

Using numerical methods (such as Newton's method or graphing), we can find approximate solutions for \( y \).

### Step 3: Back-substitute for \( x \)
Once you find \( y \), recall that \( y = \sqrt{x} \), so square the value of \( y \) to get the corresponding values for \( x \).

Would you like me to proceed with a numerical solution for this equation?

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Here are some related questions to consider:
1. How do you solve higher-degree polynomials?
2. Can all quartic equations be solved algebraically?
3. What is the process of substitution and when is it useful in algebra?
4. What numerical methods can be used to solve non-linear equations?
5. What is the importance of checking solutions when dealing with square roots?

**Tip**: Always check for extraneous solutions when solving equations that involve square roots or higher powers.

Discover how to solve the equation x^2 + sqrt(x) - 10 by using substitution and tackling a quartic equation. This solution explores advanced algebraic concepts and numerical methods suitable for Grades 11-12 students.

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