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Solve 3x² - 3x - 6 = 0

The solutions of 3x² - 3x - 6 = 0 are x = 2 or x = -1.

x² +x += 0
3x2−3x−6=0⇒x=2, −13x^2 - 3x - 6 = 0 \Rightarrow x = 2,\ -1
Discriminant D = 81 · x ≈ 2, -1

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

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Graph of y = 3x² - 3x - 6-3-2-11234-1010203040x = 2x = -1vertexy = 3x² - 3x - 6
Graph of y = 3x² - 3x - 6. The curve crosses the x-axis at x = 2 and x = -1; the vertex is at (0.5, -6.75).

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 = -3, c = -6.

    3x2−3x−6=03x^2 - 3x - 6 = 0
  2. 2

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

    D=(−3)2−4⋅3⋅(−6)=9+72=81D = (-3)^2 - 4\cdot3\cdot(-6) = 9 + 72 = 81
  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:

    3(x−2)(x+1)=03\left(x - 2\right)\left(x + 1\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−2=0⇒x=2x+1=0⇒x=−1x - 2 = 0 \Rightarrow x = 2\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±816x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{81}}{6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 2 back in: a·x² + b·x + c = 3·(4) + (-3)·(2) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 3x - 6 = 0\ \Rightarrow\ x = 2,\ -1 \]
\[ 3x^2 - 3x - 6 = 0 \]
\[ D = (-3)^2 - 4\cdot3\cdot(-6) = 9 + 72 = 81 \]
\[ 3\left(x - 2\right)\left(x + 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{81}}{6} \]
\[ x = 2\quad\text{or}\quad x = -1 \]

The parabola y = 3x² - 3x - 6

xy
-2.520.25
-1.55.25
-0.5-3.75
0.5-6.75
1.5-3.75
2.55.25
3.520.25

Questions

What are the solutions of 3x² - 3x - 6 = 0?

The solutions of 3x² - 3x - 6 = 0 are x = 2 or x = -1.

Does 3x² - 3x - 6 = 0 factor?

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

What is the vertex of y = 3x² - 3x - 6?

(1/2, -27/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -2.

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