1. **State the problem:** Solve the equation $\binom{x}{2} + 3 = 7$ for $x$. 2. **Recall the formula:** The binomial coefficient $\binom{x}{2}$ is defined as $$\binom{x}{2} = \frac{x(x-1)}{2}.$$ This represents the number of ways to choose 2 items from $x$ items. 3. **Rewrite the equation using the formula:** $$\frac{x(x-1)}{2} + 3 = 7.$$ 4. **Isolate the binomial term:** $$\frac{x(x-1)}{2} = 7 - 3 = 4.$$ 5. **Multiply both sides by 2 to clear the denominator:** $$\cancel{2} \times \frac{x(x-1)}{\cancel{2}} = 4 \times 2$$ $$x(x-1) = 8.$$ 6. **Expand and form a quadratic equation:** $$x^2 - x = 8.$$ 7. **Bring all terms to one side:** $$x^2 - x - 8 = 0.$$ 8. **Solve the quadratic equation using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=-8$. 9. **Calculate the discriminant:** $$\Delta = (-1)^2 - 4 \times 1 \times (-8) = 1 + 32 = 33.$$ 10. **Find the roots:** $$x = \frac{1 \pm \sqrt{33}}{2}.$$ 11. **Final answer:** $$x = \frac{1 + \sqrt{33}}{2} \quad \text{or} \quad x = \frac{1 - \sqrt{33}}{2}.$$ Since $x$ represents a number of items in a binomial coefficient, it is usually a non-negative integer, but here $x$ can be any real number. Both solutions are valid mathematically.