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The quadratic expression \( 3x^{2} - 6x - 9 \) can be factored as \( 3(x - 3)(x + 1) \). Solving the equation \( 3x^{2} - 6x - 9 = 0 \) gives the solutions \( x = 3 \) and \( x = -1 \).
Solution
It looks like you've provided the quadratic expression:
\[ 3x^{2} - 6x - 9 \]
Depending on what you're looking to do with this expression, here are a few common approaches:
### 1. **Factoring the Expression**
First, let's factor out the greatest common factor (GCF):
\[ 3x^{2} - 6x - 9 = 3(x^{2} - 2x - 3) \]
Now, factor the quadratic inside the parentheses:
We need two numbers that multiply to **-3** (the constant term) and add up to **-2** (the coefficient of the middle term).
The numbers **-3** and **+1** fit these requirements because:
- \(-3 \times 1 = -3\)
- \(-3 + 1 = -2\)
So, the factored form is:
\[ 3(x - 3)(x + 1) \]
### 2. **Solving the Equation \( 3x^{2} - 6x - 9 = 0 \)**
If you're looking to solve the quadratic equation, here's how you can do it:
**Step 1: Simplify the Equation**
Divide the entire equation by the GCF, which is 3:
\[ x^{2} - 2x - 3 = 0 \]
**Step 2: Factor the Simplified Equation**
As factored earlier:
\[ (x - 3)(x + 1) = 0 \]
**Step 3: Find the Solutions**
Set each factor equal to zero and solve for \( x \):
1. \( x - 3 = 0 \) ⟹ \( x = 3 \)
2. \( x + 1 = 0 \) ⟹ \( x = -1 \)
**Solutions:** \( x = 3 \) and \( x = -1 \)
### 3. **Using the Quadratic Formula**
Alternatively, you can use the quadratic formula to find the roots of the equation:
\[ ax^{2} + bx + c = 0 \]
For the equation \( 3x^{2} - 6x - 9 = 0 \):
- \( a = 3 \)
- \( b = -6 \)
- \( c = -9 \)
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
Plugging in the values:
\[ x = \frac{6 \pm \sqrt{(-6)^{2} - 4(3)(-9)}}{2(3)} \]
\[ x = \frac{6 \pm \sqrt{36 + 108}}{6} \]
\[ x = \frac{6 \pm \sqrt{144}}{6} \]
\[ x = \frac{6 \pm 12}{6} \]
So, the solutions are:
1. \( x = \frac{6 + 12}{6} = \frac{18}{6} = 3 \)
2. \( x = \frac{6 - 12}{6} = \frac{-6}{6} = -1 \)
**Solutions:** \( x = 3 \) and \( x = -1 \)
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**Summary:**
- **Factored Form:** \( 3(x - 3)(x + 1) \)
- **Solutions to \( 3x^{2} - 6x - 9 = 0 \):** \( x = 3 \) and \( x = -1 \)
If you had a different operation or question in mind regarding this expression, please let me know!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
Did you know that quadratic equations like \( 3x^{2} - 6x - 9 \) can often represent real-world phenomena such as projectile motion or the trajectory of objects? By solving these equations, we can predict outcomes like how far an object will travel or how high it will go. So next time you’re throwing a ball, remember there’s some math behind that arc! Now, when factoring quadratics like this one, keep an eye out for common factors! In this case, all terms can be simplified by 3. So you'll want to rewrite it as \( x^{2} - 2x - 3 \) first. Then, you can find two numbers that multiply to -3 and add to -2, leading you to the factors of the complete expression. Happy factoring!
