Solve 20.2x^3 - 80.8x^2 + 60.6 = 0. Give all three roots exactly and as decimals.

Step-by-step solution

1. 1

   Clear the decimals and reduce. Multiplying by $10$ gives $202x^{3}-808x^{2}+606=0$, and every coefficient is even, so divide by $2$:

   $101x^{3}-404x^{2}+303=0.$

   Note there is no $x$ term — its coefficient is $0$, which must be remembered when dividing later. Working with integers from here on makes the rational-root search exact rather than approximate.

2. 2

   Find the rational root. The coefficients are all multiples of $101$ except in a suggestive pattern: $101$, $-404=-4\cdot101$, $303=3\cdot101$. Testing $x=1$:

   $101-404+303=0 ✓,$

   so $x=1$ is a root and $(x-1)$ divides the cubic. This is the payoff of reducing first: the test is one line of mental arithmetic.

3. 3

   Divide out $(x-1)$. Synthetic division on the coefficients $101,\;-404,\;0,\;303$ with root $1$ produces $101,\;-303,\;-303$ and remainder $0$:

   $101x^{3}-404x^{2}+303=(x-1)\left(101x^{2}-303x-303\right).$

   The missing $x$ term must be entered as a $0$ in the division row; omitting it shifts every subsequent coefficient and wrecks the result.

4. 4

   Solve the quadratic and simplify the surd. For $101x^{2}-303x-303=0$,

   $\Delta=303^{2}+4(101)(303)=91809+122412=214221.$

   This does simplify: $214221=101^{2}\times 21$, so $\sqrt{214221}=101\sqrt{21}$. Hence

   $x=\frac{303\pm101\sqrt{21}}{202}=\frac{101\left(3\pm\sqrt{21}\right)}{202}=\frac{3\pm\sqrt{21}}{2}.$

   The common factor $101$ cancels completely — the whole cubic was $101$ times something simple.

5. 5

   Give decimals and check. With $\sqrt{21}\approx4.58258$,

   $x=1,\qquad x\approx 3.791288,\qquad x\approx -0.791288.$

   Substituting each into $20.2x^{3}-80.8x^{2}+60.6$ returns $0$ to within $10^{-13}$ ✓. Vieta also checks out: the roots sum to $1+3=4=\tfrac{404}{101}$ ✓, and their product is $1\cdot\frac{9-21}{4}=-3=-\tfrac{303}{101}$ ✓.