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Polynomial equationsAlgebraHigh schoolabout 1:46

Solve x^3 - 6x = 0: animated lesson

A narrated, step-by-step polynomial equations video lesson with captions. Final answer: x = - √6 ≈ -2.4494897, x = 0, x = √6 ≈ 2.4494897.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.7 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    x3−6 x=0{ x}^{3}-6~ x = 0
  2. 2. Steps, part 1
    1. x3−6 x=0{ x}^{3}-6~ x = 0
      Start with the equation
    2. x3−6x=0x^{3} - 6x = 0
      Move every term to one side
    3. x2−6=0x^{2} - 6 = 0
      Test candidates from the rational root theorem: x = 0 is a root
    4. x2−6=0x^{2} - 6 = 0
      Solve the remaining quadratic
  3. 3. Steps, part 2
    1. Δ=b2−4ac=02−4⋅1⋅(−6)=24\Delta = b^2 - 4ac = 0^2 - 4 \cdot 1 \cdot (-6) = 24
      Compute the discriminant
    2. Δ > 0, so there are two distinct real roots
    3. x=−b±Δ2a=0±242x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{0 \pm \sqrt{24}}{2}
      Apply the quadratic formula
    4. 24=26\sqrt{24} = 2\sqrt{6}
      Simplify the square root
  4. 4. Steps, part 3
    1. x=±6x = \pm \sqrt{6}
      Reduce the fraction
    2. x=−6≈−2.4494897,x=0,x=6≈2.4494897x = - \sqrt{6} \approx -2.4494897,\quad x = 0,\quad x = \sqrt{6} \approx 2.4494897
      Collect the real solutions
  5. 5. On the graph
    -3-2-1123-10-8-6-4-2246810x = 0
    Each solution is where the graph of left side minus right side crosses zero.
  6. 6. The answer
    x=−6≈−2.4494897, x=0, x=6≈2.4494897x = - \sqrt{6} \approx -2.4494897,\ x = 0,\ x = \sqrt{6} \approx 2.4494897

    x = - √6 ≈ -2.4494897, x = 0, x = √6 ≈ 2.4494897

  7. 7. Check
    • x = -2.4494897: left side = 0, right side = 0 ✓
    • x = 0: left side = 0, right side = 0 ✓
    • x = 2.4494897: left side = 0, right side = 0 ✓

Transcript

The problem. Here is the problem: x cubed minus 6x equals 0. A polynomial equation of degree 3 in x.

Steps, part 1. Start with the equation. Move every term to one side. Test candidates from the rational root theorem: x equals 0 is a root. Synthetic division by (x minus 0) leaves a polynomial of degree 2. Solve the remaining quadratic.

Steps, part 2. Compute the discriminant. delta greater than 0, so there are two distinct real roots. Apply the quadratic formula. Simplify the square root.

Steps, part 3. Reduce the fraction. Collect the real solutions.

On the graph. Now look at the graph of the left side minus the right side. It crosses zero exactly at the solutions we found.

The answer. So the answer is x equals minus the square root of 6 approximately negative 2.4494897, x equals 0, x equals the square root of 6 approximately 2.4494897. A degree-3 polynomial has exactly 3 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

Check. x equals negative 2.4494897: left side equals 0, right side equals 0, which checks out. x equals 0: left side equals 0, right side equals 0, which checks out. x equals 2.4494897: left side equals 0, right side equals 0, which checks out.

About polynomial equations

Find all roots of cubic and higher-degree polynomials using the rational root test and factoring.

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