solve the equation (2x+3)² - (x-3)² =33 using solving equations transformable to quadratic equation

To solve the equation \((2x + 3)^2 - (x - 3)^2 = 33\) by transforming it into a quadratic equation, follow these steps:

### Step 1: Expand the squares.
First, expand the terms on the left-hand side:

\[
(2x + 3)^2 - (x - 3)^2 = 33
\]

Expanding each square:

\[
(2x + 3)^2 = (2x)^2 + 2 \cdot (2x) \cdot 3 + 3^2 = 4x^2 + 12x + 9
\]

\[
(x - 3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2 = x^2 - 6x + 9
\]

So, the equation becomes:

\[
(4x^2 + 12x + 9) - (x^2 - 6x + 9) = 33
\]

### Step 2: Simplify the equation.
Distribute the subtraction:

\[
4x^2 + 12x + 9 - x^2 + 6x - 9 = 33
\]

Combine like terms:

\[
3x^2 + 18x = 33
\]

### Step 3: Rearrange into standard quadratic form.
Subtract 33 from both sides to set the equation equal to 0:

\[
3x^2 + 18x - 33 = 0
\]

### Step 4: Solve the quadratic equation.
Divide the entire equation by 3 to simplify:

\[
x^2 + 6x - 11 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \( a = 1 \), \( b = 6 \), and \( c = -11 \):

\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-11)}}{2 \cdot 1}
\]

\[
x = \frac{-6 \pm \sqrt{36 + 44}}{2}
\]

\[
x = \frac{-6 \pm \sqrt{80}}{2}
\]

\[
x = \frac{-6 \pm 4\sqrt{5}}{2}
\]

\[
x = -3 \pm 2\sqrt{5}
\]

### Final Answer:
The solutions are:

\[
x = -3 + 2\sqrt{5} \quad \text{or} \quad x = -3 - 2\sqrt{5}
\]

Would you like more details or have any questions?

### Related Practice Questions:
1. Solve \((3x - 4)^2 - (x + 2)^2 = 20\) using the quadratic equation method.
2. Solve \(x^2 + 6x - 11 = 0\) by completing the square.
3. Determine the roots of \(2x^2 + 8x - 30 = 0\) using the quadratic formula.
4. Find the value of \(x\) if \((4x + 5)^2 - (2x - 1)^2 = 48\).
5. Solve \(x^2 + 7x - 6 = 0\) by factoring.

### Tip:
When solving quadratic equations, simplifying the coefficients early (if possible) can make the calculation more straightforward.

Learn how to solve the equation (2x+3)^2 - (x-3)^2 = 33 step-by-step by transforming it into a quadratic equation. This problem involves expanding squares, simplifying expressions, and solving quadratic equations. Suitable for high school students.

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