\( 30 x^{2}+2 x-24=0 \) \( 0 x^{2}-2 x-15=0 \) \( 02 a^{2}-11 x-6=0 \) H/W \( =1 \) (d) \( x^{2}+6 x=1 \) \( x^{2}-4 x-=0 \)

Sure, I'd be happy to help you solve these quadratic equations. Let's go through each one step by step. I'll also address any apparent typos or ambiguities to ensure accurate solutions. --- ### 1. \( 30x^{2} + 2x - 24 = 0 \) **Simplify the Equation:** First, let's see if we can simplify the equation by dividing all terms by a common factor. In this case, we can divide by 2: \[ 30x^{2} + 2x - 24 = 0 \\ \Rightarrow 15x^{2} + x - 12 = 0 \] **Solve Using the Quadratic Formula:** The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \(15x^{2} + x - 12 = 0\): - \( a = 15 \) - \( b = 1 \) - \( c = -12 \) **Calculate the Discriminant (\( D \)):** \[ D = b^2 - 4ac = 1^2 - 4 \times 15 \times (-12) = 1 + 720 = 721 \] **Find the Roots:** \[ x = \frac{-1 \pm \sqrt{721}}{2 \times 15} = \frac{-1 \pm \sqrt{721}}{30} \] \[ \sqrt{721} \approx 26.85 \] **Approximate Solutions:** \[ x \approx \frac{-1 + 26.85}{30} \approx \frac{25.85}{30} \approx 0.8617 \] \[ x \approx \frac{-1 - 26.85}{30} \approx \frac{-27.85}{30} \approx -0.9283 \] **Exact Solutions:** \[ x = \frac{-1 \pm \sqrt{721}}{30} \] --- ### 2. \( 0x^{2} - 2x - 15 = 0 \) **Interpretation:** Since the coefficient of \( x^2 \) is 0, this is a linear equation, not a quadratic one. **Simplify the Equation:** \[ -2x - 15 = 0 \\ \Rightarrow -2x = 15 \\ \Rightarrow x = -\frac{15}{2} \\ \Rightarrow x = -7.5 \] **Solution:** \[ x = -7.5 \] --- ### 3. \( 02a^{2} - 11x - 6 = 0 \) (Assuming H/W \(=1\)) **Clarification Needed:** There appear to be some typos or ambiguities in this equation: - **"02a²"**: This could mean \(0 \times 2a^2 = 0\), reducing the equation to \(-11x - 6 = 0\). - **Variables Mixed**: The equation has both \( a \) and \( x \), which is unusual unless it's a system of equations or relates to homework 1. **Assuming It’s a Linear Equation in \( x \):** If we consider \(02a^{2} = 0\), then: \[ -11x - 6 = 0 \\ \Rightarrow -11x = 6 \\ \Rightarrow x = -\frac{6}{11} \] **Solution:** \[ x = -\frac{6}{11} \approx -0.5455 \] *If there’s more context or if \( a \) is involved in another part of the problem, please provide additional details.* --- ### 4. (d) \( x^{2} + 6x = 1 \) **Rewrite in Standard Quadratic Form:** \[ x^{2} + 6x - 1 = 0 \] **Solve Using the Quadratic Formula:** \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here: - \( a = 1 \) - \( b = 6 \) - \( c = -1 \) **Calculate the Discriminant (\( D \)):** \[ D = 6^2 - 4 \times 1 \times (-1) = 36 + 4 = 40 \] **Find the Roots:** \[ x = \frac{-6 \pm \sqrt{40}}{2 \times 1} = \frac{-6 \pm 2\sqrt{10}}{2} = -3 \pm \sqrt{10} \] **Approximate Solutions:** \[ \sqrt{10} \approx 3.1623 \\ x \approx -3 + 3.1623 \approx 0.1623 \\ x \approx -3 - 3.1623 \approx -6.1623 \] **Solutions:** \[ x = -3 + \sqrt{10} \quad \text{and} \quad x = -3 - \sqrt{10} \] --- ### 5. \( x^{2} - 4x - = 0 \) **Clarification Needed:** There's an incomplete term in this equation: \( x^{2} - 4x - \quad = 0 \). It seems like a value is missing after the second minus sign. I'll consider two possible interpretations: #### **a. Missing Constant Term:** If the equation is \( x^{2} - 4x = 0 \): **Factor the Equation:** \[ x(x - 4) = 0 \] **Find the Roots:** \[ x = 0 \quad \text{or} \quad x = 4 \] **Solutions:** \[ x = 0 \quad \text{and} \quad x = 4 \] #### **b. Typographical Error with a Constant:** If the equation was intended to have a constant, for example, \( x^{2} - 4x - c = 0 \), where \( c \) is a number, please provide the value of \( c \) to solve accordingly. --- **Summary of Solutions:** 1. \( 30x^{2} + 2x - 24 = 0 \) - Exact: \( x = \frac{-1 \pm \sqrt{721}}{30} \) - Approximate: \( x \approx 0.8617 \) and \( x \approx -0.9283 \) 2. \( 0x^{2} - 2x - 15 = 0 \) - \( x = -7.5 \) 3. \( 02a^{2} - 11x - 6 = 0 \) (Assuming linear in \( x \)) - \( x = -\frac{6}{11} \approx -0.5455 \) 4. \( x^{2} + 6x = 1 \) - Exact: \( x = -3 \pm \sqrt{10} \) - Approximate: \( x \approx 0.1623 \) and \( x \approx -6.1623 \) 5. \( x^{2} - 4x - = 0 \) - If \( x^{2} - 4x = 0 \): \( x = 0 \) and \( x = 4 \) --- If you have any further information or specific questions about these problems, feel free to ask!