1. **State the problem:** Find all x-intercepts of the function $$f(x) = \frac{5x^2 - 47x + 56}{2x + 8}$$. The x-intercepts occur where the function equals zero, i.e., where the numerator is zero and the denominator is not zero. 2. **Formula and rules:** To find x-intercepts, solve $$f(x) = 0$$ which means $$\frac{5x^2 - 47x + 56}{2x + 8} = 0$$. This implies the numerator must be zero: $$5x^2 - 47x + 56 = 0$$ 3. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=5$$, $$b=-47$$, and $$c=56$$. Calculate the discriminant: $$\Delta = (-47)^2 - 4 \times 5 \times 56 = 2209 - 1120 = 1089$$ 4. **Find the roots:** $$x = \frac{47 \pm \sqrt{1089}}{10} = \frac{47 \pm 33}{10}$$ 5. **Calculate each root:** - $$x_1 = \frac{47 + 33}{10} = \frac{80}{10} = 8$$ - $$x_2 = \frac{47 - 33}{10} = \frac{14}{10} = 1.4$$ 6. **Check denominator for these x-values:** $$2x + 8 \neq 0$$ For $$x=8$$: $$2(8) + 8 = 16 + 8 = 24 \neq 0$$ For $$x=1.4$$: $$2(1.4) + 8 = 2.8 + 8 = 10.8 \neq 0$$ Both values are valid x-intercepts. 7. **Write the x-intercepts as coordinate points:** $$(8, 0)$$ and $$(1.4, 0)$$ **Final answer:** The function has two x-intercepts at $$(8, 0)$$ and $$(1.4, 0)$$.