Solve the cubic equation 49 - 112x + 21x^2 - x^3 = 0.

Step-by-step solution

1. 1

   Put the cubic in monic standard form. Reversing the order and multiplying through by $-1$:

   $x^3 - 21x^2 + 112x - 49 = 0$

   A leading coefficient of $+1$ is what every root test and depression formula below assumes.

2. 2

   Rule out rational roots. By the rational root theorem the only candidates are $\pm 1, \pm 7, \pm 49$. Testing them: $f(1) = 43$, $f(-1) = -183$, $f(7) = 343 - 1029 + 784 - 49 = 49$, $f(-7) = -2205$, and $f(\pm 49)$ is far from zero. So the cubic does not factor over the rationals and no synthetic-division shortcut exists.

3. 3

   Depress the cubic by shifting away the quadratic term. For $x^3 + ax^2 + bx + c$ the substitution $x = y - \tfrac{a}{3}$ kills the square term. Here $a = -21$, so put $x = y + 7$:

   $(y+7)^3 - 21(y+7)^2 + 112(y+7) - 49 = y^3 - 35y + 49 = 0$

   The depressed form $y^3 + py + q$ has $p = -35$, $q = 49$.

4. 4

   Confirm all three roots are real, then apply the cosine formula. The cubic discriminant term is $\left(\tfrac{q}{2}\right)^2 + \left(\tfrac{p}{3}\right)^3 = 600.25 - 1587.963 = -987.713 < 0$, the casus irreducibilis: three distinct real roots that the Cardano radical formula can only express through complex numbers. The trigonometric substitution $y = 2\sqrt{-p/3}\,\cos\theta$ avoids that:

   $y_k = 2\sqrt{\frac{35}{3}}\,\cos\!\left(\frac{1}{3}\arccos\!\left(-\frac{21}{10}\sqrt{\frac{3}{35}}\right) - \frac{2\pi k}{3}\right), \quad k = 0, 1, 2$

5. 5

   Shift back and evaluate. Since $x = y + 7$ and $2\sqrt{35/3} \approx 6.8313$ with $\arccos(-0.614824) \approx 2.23207$ radians, the three roots are

   $x_0 \approx 12.0248, \qquad x_1 \approx 8.4956, \qquad x_2 \approx 0.4797$

6. 6

   Check against Vieta's formulas. The three roots must sum to $21$ and multiply to $49$: $12.0248 + 8.4956 + 0.4797 = 21.0001$ and $12.0248 \times 8.4956 \times 0.4797 = 49.00$. Both match, which the commonly quoted decimals $0.4582,\ 9.8091,\ 10.7327$ do not — those sum to $21$ but their product is $48.24$, so they are not the roots of this cubic.