1. **State the problem:** Solve the quadratic equation using the quadratic formula where $a=1$, $b=-10$, and $c=21$. 2. **Recall the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions for any quadratic equation $ax^2 + bx + c = 0$. 3. **Substitute the values:** $$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(21)}}{2(1)} = \frac{10 \pm \sqrt{100 - 84}}{2}$$ 4. **Simplify under the square root:** $$\sqrt{100 - 84} = \sqrt{16} = 4$$ 5. **Write the two possible solutions:** $$x = \frac{10 + 4}{2} \quad \text{or} \quad x = \frac{10 - 4}{2}$$ 6. **Calculate each solution:** $$x = \frac{14}{2} = 7$$ $$x = \frac{6}{2} = 3$$ 7. **Final answer:** The solutions to the quadratic equation are $x=7$ and $x=3$.