1. We are given the system of equations: $$a+b=21$$ $$ab=4$$ 2. We want to find the values of $a$ and $b$ that satisfy both equations. 3. From the first equation, express $b$ in terms of $a$: $$b=21 - a$$ 4. Substitute $b$ into the second equation: $$a(21 - a) = 4$$ 5. Expand and rearrange to form a quadratic equation: $$21a - a^2 = 4$$ $$-a^2 + 21a - 4 = 0$$ Multiply both sides by $-1$ for standard form: $$a^2 - 21a + 4 = 0$$ 6. Use the quadratic formula to solve for $a$: $$a = \frac{21 \pm \sqrt{21^2 - 4 \times 1 \times 4}}{2} = \frac{21 \pm \sqrt{441 - 16}}{2} = \frac{21 \pm \sqrt{425}}{2}$$ 7. Simplify the square root: $$\sqrt{425} = \sqrt{25 \times 17} = 5\sqrt{17}$$ 8. So the solutions for $a$ are: $$a = \frac{21 \pm 5\sqrt{17}}{2}$$ 9. Find corresponding $b$ values using $b = 21 - a$: $$b = 21 - \frac{21 \pm 5\sqrt{17}}{2} = \frac{42 - 21 \mp 5\sqrt{17}}{2} = \frac{21 \mp 5\sqrt{17}}{2}$$ 10. Therefore, the two pairs $(a,b)$ are: $$\left( \frac{21 + 5\sqrt{17}}{2}, \frac{21 - 5\sqrt{17}}{2} \right) \quad \text{and} \quad \left( \frac{21 - 5\sqrt{17}}{2}, \frac{21 + 5\sqrt{17}}{2} \right)$$ These satisfy both original equations.