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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, 23-3x^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 + 2-3-2-1123-30-25-20-15-10-55x = -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=0-3x^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\cdot(-3)\cdot2 = 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±25−6x = \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
\[ -3x^2 - x + 2 = 0\ \Rightarrow\ x = -1,\ \frac{2}{3} \]
\[ -3x^2 - x + 2 = 0 \]
\[ D = (-1)^2 - 4\cdot(-3)\cdot2 = 1 + 24 = 25 \]
\[ -\left(x + 1\right)\left(3x - 2\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{-6} \]
\[ x = -1\quad\text{or}\quad x = \frac{2}{3} \]

The parabola y = -3x² - x + 2

xy
-3.1667-24.917267
-2.1667-9.9170667
-1.1667-0.91686667
-0.16672.0833333
0.8333-0.91646667
1.8333-9.9162667
2.8333-24.916067

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

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

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

What are the sum and product of the roots?

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

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