1. **State the problem:** Solve the quadratic equation $$5x^2 = 12 - 17x$$ for $x$. 2. **Rewrite the equation in standard form:** Move all terms to one side to get zero on the other side. $$5x^2 + 17x - 12 = 0$$ 3. **Identify coefficients:** Here, $a = 5$, $b = 17$, and $c = -12$. 4. **Recall the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions for any quadratic equation $ax^2 + bx + c = 0$. 5. **Calculate the discriminant:** $$b^2 - 4ac = 17^2 - 4 \times 5 \times (-12) = 289 + 240 = 529$$ 6. **Take the square root of the discriminant:** $$\sqrt{529} = 23$$ 7. **Substitute values into the quadratic formula:** $$x = \frac{-17 \pm 23}{2 \times 5} = \frac{-17 \pm 23}{10}$$ 8. **Calculate the two possible solutions:** - For the plus sign: $$x = \frac{-17 + 23}{10} = \frac{6}{10} = \frac{3}{5}$$ - For the minus sign: $$x = \frac{-17 - 23}{10} = \frac{-40}{10} = -4$$ 9. **Final answer:** The solutions to the equation are $$x = \frac{3}{5} \quad \text{or} \quad x = -4$$