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Quadratic equationsAlgebraHigh schoolabout 1:29

Solve 3x^2 + 2x + 1 = 0: animated lesson

A narrated, step-by-step quadratic equations video lesson with captions. Final answer: x = (-1 - √2·i)/3 or x = (-1 + √2·i)/3.

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    3 x2+2 x+1=03~{ x}^{2}+2~ x+1 = 0
  2. 2. Steps, part 1
    1. 3 x2+2 x+1=03~{ x}^{2}+2~ x+1 = 0
      Start with the equation
    2. 3x2+2x+1=03x^{2} + 2x + 1 = 0
      Move every term to one side to get standard form
    3. Δ=b2−4ac=22−4⋅3⋅1=−8\Delta = b^2 - 4ac = 2^2 - 4 \cdot 3 \cdot 1 = -8
      Compute the discriminant
    4. Δ < 0, so there are no real roots; the two roots are complex conjugates
  3. 3. Steps, part 2
    1. x=−2±−86=−2±22 i6x = \frac{-2 \pm \sqrt{-8}}{6} = \frac{-2 \pm 2\sqrt{2}\,i}{6}
      Apply the quadratic formula with √(−1) = i
    2. x=−1±2 i3x = \frac{-1 \pm \sqrt{2}\,i}{3}
      Simplify
  4. 4. The answer
    x1=−1−2 i3,x2=−1+2 i3x_1 = \frac{-1 - \sqrt{2}\,i}{3},\quad x_2 = \frac{-1 + \sqrt{2}\,i}{3}

    Δ = -8; x = (-1 - √2·i)/3 or x = (-1 + √2·i)/3

  5. 5. Check
    • Vieta: sum of roots = -0.66666667 = −b/a = -0.66666667; product = 0.33333333 = c/a = 0.33333333

Transcript

The problem. Here is the problem: 3x squared plus 2x plus 1 equals 0. A quadratic equation in x: solve with the quadratic formula.

Steps, part 1. Start with the equation. Move every term to one side to get standard form. Compute the discriminant. delta less than 0, so there are no real roots; the two roots are complex conjugates.

Steps, part 2. Apply the quadratic formula with the square root of negative 1 equals i. Simplify.

The answer. So the answer is x equals (negative 1 minus the square root of 2 times i/3 or x equals (negative 1 plus the square root of 2 times i/3. The discriminant is negative, so there are no real solutions; the solutions are a pair of complex conjugates.

Check. Vieta: sum of roots equals negative 0.66666667 equals negative b/a equals negative 0.66666667; product equals 0.33333333 equals c/a equals 0.33333333.

About quadratic equations

Solve quadratics by factoring, completing the square or the quadratic formula, including complex roots.

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Solve 3x^2 + 2x + 1 = 0: Animated Step-by-Step Lesson