Solve using the quadratic formula. Round your answers to the nearest thousandth as needed. \( 5 x^{2}-12 x=4 \)

1. Write the equation in standard form: \[ 5x^2 - 12x - 4 = 0 \] 2. Identify the coefficients: \[ a = 5,\quad b = -12,\quad c = -4 \] 3. Apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the coefficients: \[ x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(5)(-4)}}{2(5)} = \frac{12 \pm \sqrt{144 + 80}}{10} \] 4. Simplify under the square root: \[ \sqrt{144+80} = \sqrt{224} \] Notice that \(224 = 16 \times 14\), so: \[ \sqrt{224} = 4\sqrt{14} \] 5. Substitute back into the formula: \[ x = \frac{12 \pm 4\sqrt{14}}{10} = \frac{6 \pm 2\sqrt{14}}{5} \] 6. Find the approximate numerical values by evaluating \(\sqrt{14} \approx 3.742\): - For the positive root: \[ x \approx \frac{6 + 2(3.742)}{5} = \frac{6 + 7.484}{5} = \frac{13.484}{5} \approx 2.697 \] - For the negative root: \[ x \approx \frac{6 - 2(3.742)}{5} = \frac{6 - 7.484}{5} = \frac{-1.484}{5} \approx -0.297 \] The solutions, rounded to the nearest thousandth, are: \[ x \approx 2.697 \quad \text{and} \quad x \approx -0.297 \]