Solve the equation \( 6 x^{2}+13 x+6=0 \) Answer: \( x=\square \)

1. **Write the equation:** \[ 6x^2+13x+6=0 \] 2. **Factor the quadratic equation:** We look for two numbers that multiply to \(6 \times 6 = 36\) and add to \(13\). The numbers \(9\) and \(4\) satisfy these conditions because \(9 \times 4 = 36\) and \(9 + 4 = 13\). 3. **Rewrite the middle term using \(9x\) and \(4x\):** \[ 6x^2+9x+4x+6=0 \] 4. **Factor by grouping:** Group the terms: \[ (6x^2+9x)+(4x+6)=0 \] Factor out the common factors in each group: - From the first group \((6x^2+9x)\), factor out \(3x\): \[ 3x(2x+3) \] - From the second group \((4x+6)\), factor out \(2\): \[ 2(2x+3) \] The equation becomes: \[ 3x(2x+3)+2(2x+3)=0 \] 5. **Factor out the common factor \((2x+3)\):** \[ (2x+3)(3x+2)=0 \] 6. **Solve for \(x\) by setting each factor equal to 0:** - Set \(2x+3=0\): \[ 2x=-3 \quad \Rightarrow \quad x=-\frac{3}{2} \] - Set \(3x+2=0\): \[ 3x=-2 \quad \Rightarrow \quad x=-\frac{2}{3} \] 7. **Final Answer:** \[ x=-\frac{3}{2} \quad \text{or} \quad x=-\frac{2}{3} \]