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Solve 3x² - x - 2 = 0

The solutions of 3x² - x - 2 = 0 are x = 1 or x = -2/3.

x² +x += 0
3x2−x−2=0⇒x=1, −233x^2 - x - 2 = 0 \Rightarrow x = 1,\ -\frac{2}{3}
Discriminant D = 25 · x ≈ 1, -0.666667

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = 3x² - x - 2-3-2-1123-551015202530x = 1x = -0.66666667vertexy = 3x² - x - 2
Graph of y = 3x² - x - 2. The curve crosses the x-axis at x = 1 and x = -0.66666667; the vertex is at (0.16666667, -2.0833333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = -1, c = -2.

    3x2−x−2=03x^2 - x - 2 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−1)2−4⋅3⋅(−2)=1+24=25D = (-1)^2 - 4\cdot3\cdot(-2) = 1 + 24 = 25
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−1)(3x+2)=0\left(x - 1\right)\left(3x + 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−1=0⇒x=13x+2=0⇒x=−23x - 1 = 0 \Rightarrow x = 1\qquad 3x + 2 = 0 \Rightarrow x = -\frac{2}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±256x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{6}
  7. 7

    Simplify to get the solutions:

    x=1orx=−23x = 1\quad\text{or}\quad x = -\frac{2}{3}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 3·(1) + (-1)·(1) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - x - 2 = 0\ \Rightarrow\ x = 1,\ -\frac{2}{3} \]
\[ 3x^2 - x - 2 = 0 \]
\[ D = (-1)^2 - 4\cdot3\cdot(-2) = 1 + 24 = 25 \]
\[ \left(x - 1\right)\left(3x + 2\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x + 2 = 0 \Rightarrow x = -\frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{6} \]
\[ x = 1\quad\text{or}\quad x = -\frac{2}{3} \]

The parabola y = 3x² - x - 2

xy
-2.833324.916067
-1.83339.9162667
-0.83330.91646667
0.1667-2.0833333
1.16670.91686667
2.16679.9170667
3.166724.917267

Questions

What are the solutions of 3x² - x - 2 = 0?

The solutions of 3x² - x - 2 = 0 are x = 1 or x = -2/3.

Does 3x² - x - 2 = 0 factor?

Yes: (x-1)(3x+2) = 0.

What is the vertex of y = 3x² - x - 2?

(1/6, -25/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1/3, product = c/a = -2/3.

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