Solve x³ − x = 0
Polynomial equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −1, x = 0, x = 1 |
Step-by-step solution
8 steps-
1 Givenx^{3} - x = 0
Solve for x.
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2 Every term contains x, so factor out x
Every term contains x, so it factors out. Setting x = 0 gives the root 0, and what is left is a polynomial of lower degree.
x \left(x^{2} - 1\right) = 0 -
3 Zero product property: a product is 0 only when one of its factors is 0
0 is the only number with this property: if a·b = 0, then a = 0 or b = 0. That is why the equation was first rearranged to … = 0 and factored: now each factor can be set to 0 on its own, giving a simpler equation for each.
x = 0\quad \text{or} \quad x^{2} - 1 = 0 -
4 This is a quadratic in standard form ax² + bx + c = 0
Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = 0 and c = -1, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = 0,\quad c = -1 -
5 There is no x term, so isolate x²x^{2} = 1
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6 Take the square root of both sides — remember both signs
A number and its negative give the same result when raised to an even power: 3² = 9 and (−3)² = 9. So if something squared equals k, that something is √k or −√k. Forgetting the minus sign loses a solution.
x = \pm 1 -
7 Solutionsx = 1\quad \text{or} \quad x = -1
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8 Check\begin{aligned}x = -1:\quad 0 = 0\quad\checkmark\\ x = 0:\quad 0 = 0\quad\checkmark\\ x = 1:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −1 | 0 | 0 | ✓ |
| 0 | 0 | 0 | ✓ |
| 1 | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.