To solve the quadratic equation \( 3x^2 + 24x = 27 \) by factoring, follow these steps:

### Step 1: Move all terms to one side of the equation
First, subtract 27 from both sides to set the equation to zero:

\[
3x^2 + 24x - 27 = 0
\]

### Step 2: Factor out the greatest common factor (GCF)
Notice that the GCF of the terms on the left-hand side is 3. Factor 3 out of the equation:

\[
3(x^2 + 8x - 9) = 0
\]

### Step 3: Factor the quadratic expression
Now, factor the quadratic expression \( x^2 + 8x - 9 \). We need to find two numbers that multiply to \(-9\) (the constant term) and add to \(8\) (the coefficient of the middle term):

However, upon further inspection, the quadratic expression \(x^2 + 8x - 9\) does not factor nicely into rational numbers. So, we'll proceed by using the quadratic formula:

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

For the equation \( x^2 + 8x - 9 = 0 \), \(a = 1\), \(b = 8\), and \(c = -9\).

### Step 4: Apply the quadratic formula
\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-9)}}{2(1)}
\]
\[
x = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
x = \frac{-8 \pm \sqrt{100}}{2}
\]
\[
x = \frac{-8 \pm 10}{2}
\]

### Step 5: Solve for the two possible values of \(x\)
\[
x = \frac{-8 + 10}{2} = \frac{2}{2} = 1
\]
\[
x = \frac{-8 - 10}{2} = \frac{-18}{2} = -9
\]

### Step 6: Write the final solution
The solutions to the equation \( 3x^2 + 24x - 27 = 0 \) are:

\[
x = 1 \quad \text{or} \quad x = -9
\]

Would you like further details or have any questions?

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1. What are the next steps after factoring a quadratic equation?
2. How do you determine the greatest common factor (GCF)?
3. What is the quadratic formula, and when is it used?
4. How do you identify when a quadratic expression cannot be factored easily?
5. Can the quadratic formula yield complex numbers as solutions?
6. How do you verify the solutions of a quadratic equation?
7. What are the advantages of factoring over using the quadratic formula?
8. What does it mean if the discriminant (inside the square root) is negative?

**Tip:** Always check whether a quadratic expression can be factored easily before moving on to the quadratic formula, as it can save time and effort.

Learn how to solve the quadratic equation 3x^2 + 24x = 27 using factoring and the quadratic formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring quadratics and using the quadratic formula. Suitable for high school students.

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