X ^ 2 + 8x = 0 5x ^ 2 - x = 0 6y ^ 2 - 30y = 0 3x ^ 2 - 1, 2x = 0 6х2-0,5х=0 1/4 * y ^ 2 + y = 0 x - 10x ^ 2 = 0 6x-0,2x²-0

Let's solve each equation step by step. 1. **Equation:** \( x^2 + 8x = 0 \) **Factoring:** \[ x(x + 8) = 0 \] **Solution:** \[ x = 0 \quad \text{or} \quad x + 8 = 0 \Rightarrow x = -8 \] **Roots:** \( x = 0, -8 \) 2. **Equation:** \( 5x^2 - x = 0 \) **Factoring:** \[ x(5x - 1) = 0 \] **Solution:** \[ x = 0 \quad \text{or} \quad 5x - 1 = 0 \Rightarrow x = \frac{1}{5} \] **Roots:** \( x = 0, \frac{1}{5} \) 3. **Equation:** \( 6y^2 - 30y = 0 \) **Factoring:** \[ 6y(y - 5) = 0 \] **Solution:** \[ y = 0 \quad \text{or} \quad y - 5 = 0 \Rightarrow y = 5 \] **Roots:** \( y = 0, 5 \) 4. **Equation:** \( 3x^2 - 1, 2x = 0 \) (assuming you intended to write \( 3x^2 - 1 + 2x = 0 \)) **Rearranging:** \[ 3x^2 + 2x - 1 = 0 \] **Using the quadratic formula:** \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} \] \[ = \frac{-2 \pm \sqrt{4 + 12}}{6} = \frac{-2 \pm \sqrt{16}}{6} = \frac{-2 \pm 4}{6} \] **Solutions:** \[ x = \frac{2}{6} = \frac{1}{3} \quad \text{and} \quad x = \frac{-6}{6} = -1 \] **Roots:** \( x = \frac{1}{3}, -1 \) 5. **Equation:** \( 6x^2 - 0.5x = 0 \) **Factoring:** \[ x(6x - 0.5) = 0 \] **Solution:** \[ x = 0 \quad \text{or} \quad 6x - 0.5 = 0 \Rightarrow x = \frac{0.5}{6} = \frac{1}{12} \] **Roots:** \( x = 0, \frac{1}{12} \) 6. **Equation:** \( \frac{1}{4} y^2 + y = 0 \) **Factoring:** \[ y(\frac{1}{4} y + 1) = 0 \] **Solution:** \[ y = 0 \quad \text{or} \quad \frac{1}{4} y + 1 = 0 \Rightarrow y = -4 \] **Roots:** \( y = 0, -4 \) 7. **Equation:** \( x - 10x^2 = 0 \) **Factoring:** \[ x(1 - 10x) = 0 \] **Solution:** \[ x = 0 \quad \text{or} \quad 1 - 10x = 0 \Rightarrow x = \frac{1}{10} \] **Roots:** \( x = 0, \frac{1}{10} \) 8. **Equation:** \( 6x - 0.2x^2 = 0 \) **Factoring:** \[ x(6 - 0.2x) = 0 \] **Solution:** \[ x = 0 \quad \text{or} \quad 6 - 0.2x = 0 \Rightarrow 0.2x = 6 \Rightarrow x = \frac{6}{0.2} = 30 \] **Roots:** \( x = 0, 30 \) ### Summary of Roots: 1. \( x = 0, -8 \) 2. \( x = 0, \frac{1}{5} \) 3. \( y = 0, 5 \) 4. \( x = \frac{1}{3}, -1 \) 5. \( x = 0, \frac{1}{12} \) 6. \( y = 0, -4 \) 7. \( x = 0, \frac{1}{10} \) 8. \( x = 0, 30 \)