Solve (515 − 0.27x)(368 − 0.27x)(98 − 0.18x) − (515 − 0.27x)·98² − (98 − 0.18x)·270² = 0 for x.

Step-by-step solution

1. 1

   Abbreviate the three linear factors. Put

   $A=515-0.27x,\qquad B=368-0.27x,\qquad C=98-0.18x,$

   so that, using $98^{2}=9604$ and $270^{2}=72900$, the equation is

   $ABC-9604A-72900C=0.$

   Naming the factors keeps the bookkeeping manageable and makes the common factor $A$ visible.

2. 2

   Expand $BC$ and group.

   $BC=(368-0.27x)(98-0.18x)=36064-92.7x+0.0486x^{2},$

   so $BC-9604=26460-92.7x+0.0486x^{2}$ and the equation becomes

   $(515-0.27x)\bigl(0.0486x^{2}-92.7x+26460\bigr)-72900(98-0.18x)=0.$

3. 3

   Expand fully with exact fractions. Working with $0.27=\tfrac{27}{100}$, $0.18=\tfrac{18}{100}$ rather than decimals avoids drift; the result is

   $-0.013122x^{3}+50.058x^{2}-41762.7x+6482700=0,$

   or, cleared of decimals by multiplying by $-\tfrac{10^{6}}{1}$ and dividing by $6561$,

   $27x^{3}-103000x^{2}+\tfrac{2320150000}{27}x-\ldots$

   More usefully, multiplying the original by $-\tfrac{500000}{6561}$ gives the monic cubic $x^{3}-\tfrac{103000}{27}x^{2}+\tfrac{2320150000}{729}x-\tfrac{120050000000}{243}=0$.

4. 4

   Find the one exact root. Testing values that make a bracket round, $x=\tfrac{24500}{27}$ makes $0.27x=245$ exactly, and substituting it into the cubic with exact fractions gives $0$. So $\left(x-\tfrac{24500}{27}\right)$ is a factor and $x=\tfrac{24500}{27}=907.\overline{407}$ is a root.

5. 5

   Divide out and solve the quadratic. Synthetic division leaves $x^{2}-\tfrac{78500}{27}x+\tfrac{4900000}{9}=0$, i.e.

   $27x^{2}-78500x+14700000=0,$

   with discriminant $78500^{2}-4(27)(14700000)=4574650000$. Since $4574650000=10^{4}\cdot 457465$ and $457465=5\cdot 91493$ is square-free, the roots stay irrational:

   $x=\frac{39250\pm 50\sqrt{457465}}{27}.$

6. 6

   State the three roots and verify. Numerically $x\approx 201.1822413$, $x=907.4074074$ and $x\approx 2706.2251661$. Substituting each back into the original product form gives $0$, $-9\times10^{-10}$ and $0$ respectively — the tiny residual is floating-point noise on a value of order $10^{6}$. A careless expansion that produces $-41777.7x+6479700$ instead is off: that cubic has no root at $24500/27$.