13) \( 3 x^{2}-6 x-9 \)

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 \( x = -1 \) --- **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!