Solve x³ + 3x² − 2x − 6 = 0

Polynomial equation, worked out line by line the way a teacher would write it.

Answer

Solutions	x = −3, x = −√2, x = √2

9 steps

1 Given

x^{3} + 3 x^{2} - 2 x - 6 = 0

Solve for x.
2 Rational root theorem: any rational root is ±(divisor of 6)/(divisor of 1)

x \in \{ -1,\; 1,\; -2,\; 2,\; -3,\; 3,\; -6,\; 6 \}
3 Test the candidates: x = -3 makes the polynomial 0, so (x + 3) is a factor

3 \left(-3\right)^{2} - 6 - 2 \left(-3\right) + \left(-3\right)^{3} = 0
4 Divide the polynomial by (x + 3) (synthetic division)

\left(x + 3\right) \left(x^{2} - 2\right) = 0
5 This is a quadratic in standard form ax² + bx + c = 0

a = 1,\quad b = 0,\quad c = -2
6 There is no x term, so isolate x²

x^{2} = 2
7 Take the square root of both sides — remember both signs

x = \pm \sqrt{2}
8 Solutions

x = \sqrt{2}\quad \text{or} \quad x = - \sqrt{2}
9 Check

\begin{aligned}x = -3:\quad 0 = 0\quad\checkmark\\ x = - \sqrt{2}:\quad 0 = 0\quad\checkmark\\ x = \sqrt{2}:\quad 0 = 0\quad\checkmark\end{aligned}

Substituting each solution back makes both sides equal.

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