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Solve -3x² - 9x - 1 = 0

The solutions of -3x² - 9x - 1 = 0 are x = (-9 + √69)/6 or x = (-9 - √69)/6, that is x ≈ -2.8844373 and x ≈ -0.11556269.

x² +x += 0
−3x2−9x−1=0⇒x=−9+696, −9−696-3x^2 - 9x - 1 = 0 \Rightarrow x = \frac{-9 + \sqrt{69}}{6},\ \frac{-9 - \sqrt{69}}{6}
Discriminant D = 69 · x ≈ -2.88444, -0.115563

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

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Graph of y = -3x² - 9x - 1-5-4-3-2-112-40-30-20-1010x = -2.8844373x = -0.11556269vertexy = -3x² - 9x - 1
Graph of y = -3x² - 9x - 1. The curve crosses the x-axis at x = -2.8844373 and x = -0.11556269; the vertex is at (-1.5, 5.75).

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 = -1.

    −3x2−9x−1=0-3x^2 - 9x - 1 = 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)⋅(−1)=81−12=69D = (-9)^2 - 4\cdot(-3)\cdot(-1) = 81 - 12 = 69
  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±69−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{69}}{-6}
  5. 5

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

    x=−9+696orx=−9−696x = \frac{-9 + \sqrt{69}}{6}\quad\text{or}\quad x = \frac{-9 - \sqrt{69}}{6}
  6. 6

    As decimals: x ≈ -2.8844373 and x ≈ -0.11556269.

  7. 7

    Check numerically: for x ≈ -2.8844373 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 9x - 1 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{69}}{6},\ \frac{-9 - \sqrt{69}}{6} \]
\[ -3x^2 - 9x - 1 = 0 \]
\[ D = (-9)^2 - 4\cdot(-3)\cdot(-1) = 81 - 12 = 69 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{69}}{-6} \]
\[ x = \frac{-9 + \sqrt{69}}{6}\quad\text{or}\quad x = \frac{-9 - \sqrt{69}}{6} \]

The parabola y = -3x² - 9x - 1

xy
-4.5-21.25
-3.5-6.25
-2.52.75
-1.55.75
-0.52.75
0.5-6.25
1.5-21.25

Questions

What are the solutions of -3x² - 9x - 1 = 0?

The solutions of -3x² - 9x - 1 = 0 are x = (-9 + √69)/6 or x = (-9 - √69)/6, that is x ≈ -2.8844373 and x ≈ -0.11556269.

Does -3x² - 9x - 1 = 0 factor?

Not over the integers, because the discriminant 69 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² - 9x - 1?

(-3/2, 23/4). 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 = 1/3.

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    Solve -3x² - 9x - 1 = 0: x = (-9 + √69)/6 or x = (-9 - √69)/6 (steps) | WordGPT