1. **State the problem:** Solve the equation $$8x^2 - 7x + 9 = 2x^2 + 6x + 7$$ for $x$. 2. **Bring all terms to one side:** Subtract $2x^2 + 6x + 7$ from both sides to set the equation to zero: $$8x^2 - 7x + 9 - (2x^2 + 6x + 7) = 0$$ 3. **Simplify inside the parentheses:** $$8x^2 - 7x + 9 - 2x^2 - 6x - 7 = 0$$ 4. **Combine like terms:** $$ (8x^2 - 2x^2) + (-7x - 6x) + (9 - 7) = 0$$ $$6x^2 - 13x + 2 = 0$$ 5. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, the solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=6$, $b=-13$, and $c=2$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-13)^2 - 4 \times 6 \times 2 = 169 - 48 = 121$$ 7. **Find the square root of the discriminant:** $$\sqrt{121} = 11$$ 8. **Calculate the two solutions:** $$x = \frac{-(-13) \pm 11}{2 \times 6} = \frac{13 \pm 11}{12}$$ 9. **First solution:** $$x = \frac{13 + 11}{12} = \frac{24}{12} = 2$$ 10. **Second solution:** $$x = \frac{13 - 11}{12} = \frac{2}{12} = \frac{1}{6}$$ **Final answer:** $$x = 2 \quad \text{or} \quad x = \frac{1}{6}$$