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Solve -3x² - 3x + 8 = 0

The solutions of -3x² - 3x + 8 = 0 are x = (-3 + √105)/6 or x = (-3 - √105)/6, that is x ≈ -2.2078251 and x ≈ 1.2078251.

x² +x += 0
−3x2−3x+8=0⇒x=−3+1056, −3−1056-3x^2 - 3x + 8 = 0 \Rightarrow x = \frac{-3 + \sqrt{105}}{6},\ \frac{-3 - \sqrt{105}}{6}
Discriminant D = 105 · x ≈ -2.20783, 1.20783

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

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Graph of y = -3x² - 3x + 8-4-3-2-1123-40-30-20-1010x = -2.2078251x = 1.2078251vertexy = -3x² - 3x + 8
Graph of y = -3x² - 3x + 8. The curve crosses the x-axis at x = -2.2078251 and x = 1.2078251; the vertex is at (-0.5, 8.75).

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 = 8.

    −3x2−3x+8=0-3x^2 - 3x + 8 = 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)⋅8=9+96=105D = (-3)^2 - 4\cdot(-3)\cdot8 = 9 + 96 = 105
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±105−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{105}}{-6}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−3+1056orx=−3−1056x = \frac{-3 + \sqrt{105}}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{105}}{6}
  6. 6

    As decimals: x ≈ -2.2078251 and x ≈ 1.2078251.

  7. 7

    Check numerically: for x ≈ -2.2078251 the left side is 8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 3x + 8 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{105}}{6},\ \frac{-3 - \sqrt{105}}{6} \]
\[ -3x^2 - 3x + 8 = 0 \]
\[ D = (-3)^2 - 4\cdot(-3)\cdot8 = 9 + 96 = 105 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{105}}{-6} \]
\[ x = \frac{-3 + \sqrt{105}}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{105}}{6} \]

The parabola y = -3x² - 3x + 8

xy
-3.5-18.25
-2.5-3.25
-1.55.75
-0.58.75
0.55.75
1.5-3.25
2.5-18.25

Questions

What are the solutions of -3x² - 3x + 8 = 0?

The solutions of -3x² - 3x + 8 = 0 are x = (-3 + √105)/6 or x = (-3 - √105)/6, that is x ≈ -2.2078251 and x ≈ 1.2078251.

Does -3x² - 3x + 8 = 0 factor?

Not over the integers, because the discriminant 105 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² - 3x + 8?

(-1/2, 35/4). 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 = -8/3.

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    Solve -3x² - 3x + 8 = 0: x = (-3 + √105)/6 or x = (-3 - √105)/6 (steps) | WordGPT