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Solve -3x² - 9x + 2 = 0

The solutions of -3x² - 9x + 2 = 0 are x = (-9 + √105)/6 or x = (-9 - √105)/6, that is x ≈ -3.2078251 and x ≈ 0.20782513.

x² +x += 0
−3x2−9x+2=0⇒x=−9+1056, −9−1056-3x^2 - 9x + 2 = 0 \Rightarrow x = \frac{-9 + \sqrt{105}}{6},\ \frac{-9 - \sqrt{105}}{6}
Discriminant D = 105 · x ≈ -3.20783, 0.207825

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

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Graph of y = -3x² - 9x + 2-5-4-3-2-112-40-30-20-1010x = -3.2078251x = 0.20782513vertexy = -3x² - 9x + 2
Graph of y = -3x² - 9x + 2. The curve crosses the x-axis at x = -3.2078251 and x = 0.20782513; the vertex is at (-1.5, 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 = -9, c = 2.

    −3x2−9x+2=0-3x^2 - 9x + 2 = 0
  2. 2

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

    D=(−9)2−4⋅(−3)⋅2=81+24=105D = (-9)^2 - 4\cdot(-3)\cdot2 = 81 + 24 = 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=9±105−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{105}}{-6}
  5. 5

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

    x=−9+1056orx=−9−1056x = \frac{-9 + \sqrt{105}}{6}\quad\text{or}\quad x = \frac{-9 - \sqrt{105}}{6}
  6. 6

    As decimals: x ≈ -3.2078251 and x ≈ 0.20782513.

  7. 7

    Check numerically: for x ≈ -3.2078251 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 9x + 2 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{105}}{6},\ \frac{-9 - \sqrt{105}}{6} \]
\[ -3x^2 - 9x + 2 = 0 \]
\[ D = (-9)^2 - 4\cdot(-3)\cdot2 = 81 + 24 = 105 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{105}}{-6} \]
\[ x = \frac{-9 + \sqrt{105}}{6}\quad\text{or}\quad x = \frac{-9 - \sqrt{105}}{6} \]

The parabola y = -3x² - 9x + 2

xy
-4.5-18.25
-3.5-3.25
-2.55.75
-1.58.75
-0.55.75
0.5-3.25
1.5-18.25

Questions

What are the solutions of -3x² - 9x + 2 = 0?

The solutions of -3x² - 9x + 2 = 0 are x = (-9 + √105)/6 or x = (-9 - √105)/6, that is x ≈ -3.2078251 and x ≈ 0.20782513.

Does -3x² - 9x + 2 = 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 - 9x + 2?

(-3/2, 35/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

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

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    Solve -3x² - 9x + 2 = 0: x = (-9 + √105)/6 or x = (-9 - √105)/6 (steps) | WordGPT