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Solve -3x² - 3x + 5 = 0

The solutions of -3x² - 3x + 5 = 0 are x = (-3 + √69)/6 or x = (-3 - √69)/6, that is x ≈ -1.8844373 and x ≈ 0.88443731.

x² +x += 0
−3x2−3x+5=0⇒x=−3+696, −3−696-3x^2 - 3x + 5 = 0 \Rightarrow x = \frac{-3 + \sqrt{69}}{6},\ \frac{-3 - \sqrt{69}}{6}
Discriminant D = 69 · x ≈ -1.88444, 0.884437

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

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Graph of y = -3x² - 3x + 5-4-3-2-1123-40-30-20-1010x = -1.8844373x = 0.88443731vertexy = -3x² - 3x + 5
Graph of y = -3x² - 3x + 5. The curve crosses the x-axis at x = -1.8844373 and x = 0.88443731; the vertex is at (-0.5, 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 = -3, c = 5.

    −3x2−3x+5=0-3x^2 - 3x + 5 = 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)⋅5=9+60=69D = (-3)^2 - 4\cdot(-3)\cdot5 = 9 + 60 = 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=3±69−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{69}}{-6}
  5. 5

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

    x=−3+696orx=−3−696x = \frac{-3 + \sqrt{69}}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{69}}{6}
  6. 6

    As decimals: x ≈ -1.8844373 and x ≈ 0.88443731.

  7. 7

    Check numerically: for x ≈ -1.8844373 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 3x + 5 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{69}}{6},\ \frac{-3 - \sqrt{69}}{6} \]
\[ -3x^2 - 3x + 5 = 0 \]
\[ D = (-3)^2 - 4\cdot(-3)\cdot5 = 9 + 60 = 69 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{69}}{-6} \]
\[ x = \frac{-3 + \sqrt{69}}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{69}}{6} \]

The parabola y = -3x² - 3x + 5

xy
-3.5-21.25
-2.5-6.25
-1.52.75
-0.55.75
0.52.75
1.5-6.25
2.5-21.25

Questions

What are the solutions of -3x² - 3x + 5 = 0?

The solutions of -3x² - 3x + 5 = 0 are x = (-3 + √69)/6 or x = (-3 - √69)/6, that is x ≈ -1.8844373 and x ≈ 0.88443731.

Does -3x² - 3x + 5 = 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 - 3x + 5?

(-1/2, 23/4). 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 = -5/3.

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    Solve -3x² - 3x + 5 = 0: x = (-3 + √69)/6 or x = (-3 - √69)/6 (steps) | WordGPT