1. **State the problem:** Solve the quadratic equation $x^2 - 7x - 34 = 10$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 7x - 34 - 10 = 0$$ which simplifies to $$x^2 - 7x - 44 = 0$$ 3. **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 $a=1$, $b=-7$, and $c=-44$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4 \times 1 \times (-44) = 49 + 176 = 225$$ 5. **Find the square root of the discriminant:** $$\sqrt{225} = 15$$ 6. **Substitute values into the quadratic formula:** $$x = \frac{-(-7) \pm 15}{2 \times 1} = \frac{7 \pm 15}{2}$$ 7. **Calculate the two possible solutions:** - For the plus sign: $$x = \frac{7 + 15}{2} = \frac{22}{2} = 11$$ - For the minus sign: $$x = \frac{7 - 15}{2} = \frac{\cancel{7 - 15}}{2} = \frac{-8}{2} = -4$$ 8. **Final answer:** The solutions to the equation are $$x = 11 \text{ or } x = -4$$