1. **State the problem:** Solve the quadratic equation $$x^2 - 7x + 10 = 23$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^2 - 7x + 10 - 23 = 0$$ 3. **Simplify:** $$x^2 - 7x - 13 = 0$$ 4. **Use the quadratic formula:** The quadratic formula for $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-7$$, and $$c=-13$$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4(1)(-13) = 49 + 52 = 101$$ 6. **Substitute values into the formula:** $$x = \frac{-(-7) \pm \sqrt{101}}{2(1)} = \frac{7 \pm \sqrt{101}}{2}$$ 7. **Final answer:** $$x = \frac{7 + \sqrt{101}}{2} \quad \text{or} \quad x = \frac{7 - \sqrt{101}}{2}$$ These are the two solutions to the quadratic equation.