1. **State the problem:** Solve the quadratic equation $$7x^2 + 17x + 52 = 6x^2 + 35x - 28$$. 2. **Bring all terms to one side:** Subtract $$6x^2 + 35x - 28$$ from both sides to set the equation to zero: $$7x^2 + 17x + 52 - (6x^2 + 35x - 28) = 0$$ 3. **Simplify the expression:** $$7x^2 + 17x + 52 - 6x^2 - 35x + 28 = 0$$ Combine like terms: $$ (7x^2 - 6x^2) + (17x - 35x) + (52 + 28) = 0$$ $$x^2 - 18x + 80 = 0$$ 4. **Use the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a=1$$, $$b=-18$$, and $$c=80$$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-18)^2 - 4 \times 1 \times 80 = 324 - 320 = 4$$ 6. **Find the roots:** $$x = \frac{-(-18) \pm \sqrt{4}}{2 \times 1} = \frac{18 \pm 2}{2}$$ 7. **Calculate each root:** - $$x_1 = \frac{18 + 2}{2} = \frac{20}{2} = 10$$ - $$x_2 = \frac{18 - 2}{2} = \frac{16}{2} = 8$$ **Final answer:** $$x = 10$$ or $$x = 8$$.