1. **Problem statement:** Find the value of $x$ for the triangle with area 30 ft², base $(x+4)$ ft, and height $x$ ft. 2. **Formula:** Area of a triangle is given by $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Set up the equation:** $$30 = \frac{1}{2} \times (x+4) \times x$$ 4. **Multiply both sides by 2 to eliminate the fraction:** $$2 \times 30 = (x+4) \times x$$ $$60 = x(x+4)$$ 5. **Expand the right side:** $$60 = x^2 + 4x$$ 6. **Rewrite as a quadratic equation:** $$x^2 + 4x - 60 = 0$$ 7. **Solve the quadratic using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=4$, $c=-60$ 8. **Calculate the discriminant:** $$\sqrt{4^2 - 4 \times 1 \times (-60)} = \sqrt{16 + 240} = \sqrt{256} = 16$$ 9. **Find the roots:** $$x = \frac{-4 \pm 16}{2}$$ 10. **Calculate each root:** - $$x = \frac{-4 + 16}{2} = \frac{12}{2} = 6$$ - $$x = \frac{-4 - 16}{2} = \frac{-20}{2} = -10$$ 11. **Interpretation:** Since $x$ represents a length, it must be positive. So, $$x = 6$$ ft. **Final answer:** $$\boxed{6}$$