1. **State the problem:** We are given two functions: $$S(x) = \frac{525}{x}$$ $$f(x) = \frac{525}{x+8}$$ and the equation: $$f(x) + 2 = S(x)$$ We need to find the value(s) of $x$ that satisfy this equation. 2. **Write the equation explicitly:** $$\frac{525}{x+8} + 2 = \frac{525}{x}$$ 3. **Isolate terms and clear denominators:** Multiply both sides by $x(x+8)$ to eliminate fractions: $$525x + 2x(x+8)(x+8) = 525(x+8)$$ But let's do it step-by-step more carefully: Multiply both sides by $x(x+8)$: $$x(x+8) \left( \frac{525}{x+8} + 2 \right) = x(x+8) \cdot \frac{525}{x}$$ This simplifies to: $$525x + 2x(x+8)(x+8) = 525(x+8)$$ 4. **Simplify the terms:** - Left side: $$525x + 2x(x+8)^2$$ - Right side: $$525x + 4200$$ 5. **Subtract $525x$ from both sides:** $$2x(x+8)^2 = 4200$$ 6. **Expand $(x+8)^2$:** $$(x+8)^2 = x^2 + 16x + 64$$ So: $$2x(x^2 + 16x + 64) = 4200$$ 7. **Distribute $2x$:** $$2x^3 + 32x^2 + 128x = 4200$$ 8. **Bring all terms to one side:** $$2x^3 + 32x^2 + 128x - 4200 = 0$$ 9. **Divide entire equation by 2 to simplify:** $$x^3 + 16x^2 + 64x - 2100 = 0$$ 10. **Solve the cubic equation:** We look for rational roots using the Rational Root Theorem. Possible roots are factors of 2100. Try $x=10$: $$10^3 + 16(10)^2 + 64(10) - 2100 = 1000 + 1600 + 640 - 2100 = 1140 \neq 0$$ Try $x=5$: $$125 + 16(25) + 320 - 2100 = 125 + 400 + 320 - 2100 = -1255 \neq 0$$ Try $x=15$: $$3375 + 16(225) + 960 - 2100 = 3375 + 3600 + 960 - 2100 = 7835 \neq 0$$ Try $x= -25$: $$-15625 + 16(625) - 1600 - 2100 = -15625 + 10000 - 1600 - 2100 = -6325 \neq 0$$ Try $x= 20$: $$8000 + 16(400) + 1280 - 2100 = 8000 + 6400 + 1280 - 2100 = 11580 \neq 0$$ Try $x= -10$: $$-1000 + 16(100) - 640 - 2100 = -1000 + 1600 - 640 - 2100 = -2140 \neq 0$$ Try $x= 7$: $$343 + 16(49) + 448 - 2100 = 343 + 784 + 448 - 2100 = -525 \neq 0$$ Try $x= 12$: $$1728 + 16(144) + 768 - 2100 = 1728 + 2304 + 768 - 2100 = 2700 \neq 0$$ Try $x= 6$: $$216 + 16(36) + 384 - 2100 = 216 + 576 + 384 - 2100 = -924 \neq 0$$ Try $x= 3$: $$27 + 16(9) + 192 - 2100 = 27 + 144 + 192 - 2100 = -1737 \neq 0$$ Try $x= 1$: $$1 + 16 + 64 - 2100 = -2019 \neq 0$$ Try $x= -4$: $$-64 + 16(16) - 256 - 2100 = -64 + 256 - 256 - 2100 = -2164 \neq 0$$ Try $x= -5$: $$-125 + 16(25) - 320 - 2100 = -125 + 400 - 320 - 2100 = -2145 \neq 0$$ Try $x= 14$: $$2744 + 16(196) + 896 - 2100 = 2744 + 3136 + 896 - 2100 = 5676 \neq 0$$ Try $x= -7$: $$-343 + 16(49) - 448 - 2100 = -343 + 784 - 448 - 2100 = -2107 \neq 0$$ Try $x= 21$: $$9261 + 16(441) + 1344 - 2100 = 9261 + 7056 + 1344 - 2100 = 14561 \neq 0$$ Try $x= -3$: $$-27 + 16(9) - 192 - 2100 = -27 + 144 - 192 - 2100 = -2175 \neq 0$$ Try $x= 25$: $$15625 + 16(625) + 1600 - 2100 = 15625 + 10000 + 1600 - 2100 = 25125 \neq 0$$ Try $x= -1$: $$-1 + 16 - 64 - 2100 = -2149 \neq 0$$ Try $x= -2$: $$-8 + 16(4) - 128 - 2100 = -8 + 64 - 128 - 2100 = -2172 \neq 0$$ Try $x= -6$: $$-216 + 16(36) - 384 - 2100 = -216 + 576 - 384 - 2100 = -2124 \neq 0$$ Try $x= 4$: $$64 + 16(16) + 256 - 2100 = 64 + 256 + 256 - 2100 = -1524 \neq 0$$ Try $x= 8$: $$512 + 16(64) + 512 - 2100 = 512 + 1024 + 512 - 2100 = -52 \neq 0$$ Try $x= 9$: $$729 + 16(81) + 576 - 2100 = 729 + 1296 + 576 - 2100 = 501 \neq 0$$ Try $x= 11$: $$1331 + 16(121) + 704 - 2100 = 1331 + 1936 + 704 - 2100 = 2871 \neq 0$$ Try $x= 13$: $$2197 + 16(169) + 832 - 2100 = 2197 + 2704 + 832 - 2100 = 4633 \neq 0$$ Try $x= -8$: $$-512 + 16(64) - 512 - 2100 = -512 + 1024 - 512 - 2100 = -2100 \neq 0$$ Try $x= -9$: $$-729 + 16(81) - 576 - 2100 = -729 + 1296 - 576 - 2100 = -2109 \neq 0$$ Try $x= -12$: $$-1728 + 16(144) - 768 - 2100 = -1728 + 2304 - 768 - 2100 = -2292 \neq 0$$ Try $x= -15$: $$-3375 + 16(225) - 960 - 2100 = -3375 + 3600 - 960 - 2100 = -2835 \neq 0$$ Try $x= -20$: $$-8000 + 16(400) - 1280 - 2100 = -8000 + 6400 - 1280 - 2100 = -4980 \neq 0$$ Try $x= -25$: $$-15625 + 16(625) - 1600 - 2100 = -15625 + 10000 - 1600 - 2100 = -6325 \neq 0$$ Since no simple rational root is found, the cubic must be solved numerically or by approximation. 11. **Numerical approximation:** Using a calculator or software, approximate the root of $$x^3 + 16x^2 + 64x - 2100 = 0$$ The approximate real root is around $x \approx 7.5$. **Final answer:** $$x \approx 7.5$$