1. **State the problem:** Solve the quadratic equation $8x^2 - 7x + 9 = 2x^2 + 6x + 7$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$8x^2 - 7x + 9 - 2x^2 - 6x - 7 = 0$$ 3. **Simplify the equation:** Combine like terms: $$ (8x^2 - 2x^2) + (-7x - 6x) + (9 - 7) = 0$$ $$6x^2 - 13x + 2 = 0$$ 4. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=6$, $b=-13$, and $c=2$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-13)^2 - 4 \times 6 \times 2 = 169 - 48 = 121$$ 6. **Find the square root of the discriminant:** $$\sqrt{121} = 11$$ 7. **Calculate the two solutions:** $$x = \frac{-(-13) \pm 11}{2 \times 6} = \frac{13 \pm 11}{12}$$ 8. **Evaluate each solution:** - For the plus sign: $$x = \frac{13 + 11}{12} = \frac{24}{12} = 2$$ - For the minus sign: $$x = \frac{13 - 11}{12} = \frac{2}{12} = \frac{1}{6}$$ 9. **Final answer:** The solutions to the equation are $$x = 2 \quad \text{and} \quad x = \frac{1}{6}$$