show normal work of a highschooler for each of the problems step by step and detailed please!

Absolutely! I'll walk you through some example problems that a high school student might encounter, showing each step in detail. Let's start with a variety of topics, including algebra, geometry, and calculus. I'll provide the work as if a high school student were solving the problems.

### Example Problem Set

1. **Algebra: Solving a Quadratic Equation**
   Solve the quadratic equation: \( x^2 - 5x + 6 = 0 \)

   **Step-by-step Solution:**

   - **Step 1: Write down the quadratic equation.**
     \[
     x^2 - 5x + 6 = 0
     \]

   - **Step 2: Identify the coefficients \(a\), \(b\), and \(c\).**
     In this equation, \(a = 1\), \(b = -5\), and \(c = 6\).

   - **Step 3: Factor the quadratic equation.**
     Look for two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)).
     These numbers are \(-2\) and \(-3\).

     So, we can factor the quadratic as:
     \[
     (x - 2)(x - 3) = 0
     \]

   - **Step 4: Set each factor equal to zero and solve for \(x\).**
     \[
     x - 2 = 0 \quad \Rightarrow \quad x = 2
     \]
     \[
     x - 3 = 0 \quad \Rightarrow \quad x = 3
     \]

   - **Step 5: Write down the solutions.**
     The solutions are \(x = 2\) and \(x = 3\).

2. **Geometry: Finding the Area of a Circle**
   Find the area of a circle with a radius of 7 cm.

   **Step-by-step Solution:**

   - **Step 1: Write down the formula for the area of a circle.**
     \[
     \text{Area} = \pi r^2
     \]

   - **Step 2: Substitute the radius value into the formula.**
     Here, \(r = 7\) cm.
     \[
     \text{Area} = \pi (7)^2
     \]

   - **Step 3: Calculate the square of the radius.**
     \[
     7^2 = 49
     \]

   - **Step 4: Multiply by \(\pi\).**
     \[
     \text{Area} = 49\pi \, \text{cm}^2
     \]

   - **Step 5: Approximate using \(\pi \approx 3.14\).**
     \[
     \text{Area} \approx 49 \times 3.14 = 153.86 \, \text{cm}^2
     \]

   - **Step 6: Write down the answer.**
     The area of the circle is approximately \(153.86 \, \text{cm}^2\).

3. **Trigonometry: Finding the Value of a Sine Function**
   Find \(\sin(30^\circ)\).

   **Step-by-step Solution:**

   - **Step 1: Recall the sine value for common angles.**
     From the unit circle or trigonometric table, we know:
     \[
     \sin(30^\circ) = \frac{1}{2}
     \]

   - **Step 2: Write down the answer.**
     \[
     \sin(30^\circ) = \frac{1}{2}
     \]

4. **Calculus: Finding the Derivative of a Function**
   Find the derivative of \( f(x) = 3x^2 + 5x - 

Explore detailed step-by-step solutions to high school level math problems in algebra, geometry, trigonometry, and calculus. Includes examples like solving quadratic equations, finding circle areas, evaluating sine functions, and calculating derivatives.

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