To solve the equation \((6x + 1)^2 + 3 = 10\), let's follow these steps:

### Step 1: Simplify the equation
Subtract 3 from both sides to isolate the squared term:

\[
(6x + 1)^2 = 10 - 3
\]

\[
(6x + 1)^2 = 7
\]

### Step 2: Take the square root of both sides
To solve for \(6x + 1\), take the square root of both sides:

\[
6x + 1 = \pm\sqrt{7}
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\) by isolating it on one side:

\[
6x = -1 \pm \sqrt{7}
\]

\[
x = \frac{-1 \pm \sqrt{7}}{6}
\]

### Final Answer
The solutions for \(x\) are:

\[
x = \frac{-1 + \sqrt{7}}{6} \quad \text{and} \quad x = \frac{-1 - \sqrt{7}}{6}
\]

Would you like further details, or do you have any questions?

### Related Questions
1. How do you solve equations involving absolute values?
2. What are the properties of square roots in solving equations?
3. How can you simplify expressions with nested radicals?
4. What methods exist for solving quadratic equations?
5. How do transformations affect the graph of a function?

**Tip:** Always check your solutions by plugging them back into the original equation to ensure accuracy!

Learn how to solve the quadratic equation (6x + 1)^2 + 3 = 10 using the square root method. This detailed solution involves algebraic techniques such as isolating terms and taking square roots. Suitable for students in Grades 9-11.

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