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Solve -3x² + 3x + 6 = 0

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

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

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

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Graph of y = -3x² + 3x + 6-3-2-11234-40-30-20-1010x = -1x = 2vertexy = -3x² + 3x + 6
Graph of y = -3x² + 3x + 6. The curve crosses the x-axis at x = -1 and x = 2; 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=0-3x^2 + 3x + 6 = 0
  2. 2

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

    D=32−4⋅(−3)⋅6=9+72=81D = 3^2 - 4\cdot(-3)\cdot6 = 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+1)(x−2)=0-3\left(x + 1\right)\left(x - 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=−1x−2=0⇒x=2x + 1 = 0 \Rightarrow x = -1\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = -1 back in: a·x² + b·x + c = -3·(1) + 3·(-1) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 3x + 6 = 0\ \Rightarrow\ x = -1,\ 2 \]
\[ -3x^2 + 3x + 6 = 0 \]
\[ D = 3^2 - 4\cdot(-3)\cdot6 = 9 + 72 = 81 \]
\[ -3\left(x + 1\right)\left(x - 2\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{81}}{-6} \]
\[ x = -1\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 3x + 6

xy
-2.5-20.25
-1.5-5.25
-0.53.75
0.56.75
1.53.75
2.5-5.25
3.5-20.25

Questions

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

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

Does -3x² + 3x + 6 = 0 factor?

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

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

(1/2, 27/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -2.

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