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Polynomial equationsAlgebraCollegeabout 1:04

Solve 2x^4 - 3x^3 + 6x^2 - 5x + 6 = 0: animated lesson

A narrated, step-by-step polynomial equations video lesson with captions. Final answer: No real solutions; complex: 0.88194688 ± 1.0018913i, -0.13194688 ± 1.2909121i.

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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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Solving an equation
$2~{ x}^{4}-3~{ x}^{3}+6~{ x}^{2}-5~ x+6 = 0$
2. Step by step
1. $2~{ x}^{4}-3~{ x}^{3}+6~{ x}^{2}-5~ x+6 = 0$
  Start with the equation
2. $2x^{4} - 3x^{3} + 6x^{2} - 5x + 6 = 0$
  Move every term to one side
3. No further rational roots; approximate the remaining 4 roots numerically
4. $\text{No real solutions}$
  Collect the real solutions
3. The answer
$\text{No real solutions}$
No real solutions

Transcript

The problem. Here is the problem: 2x to the power 4 minus 3x cubed plus 6x squared minus 5x plus 6 equals 0. A polynomial equation of degree 4 in x.

Step by step. Start with the equation. Move every term to one side. No further rational roots; approximate the remaining 4 roots numerically. The Durand–Kerner method iterates on all roots at once until they stop changing. Collect the real solutions.

The answer. So the answer is No real solutions; complex: 0.88194688 plus or minus 1.0018913i, negative 0.13194688 plus or minus 1.2909121i. A degree-4 polynomial has exactly 4 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

About polynomial equations

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

Solve 2x^4 - 3x^3 + 6x^2 - 5x + 6 = 0: Step-by-Step Lesson