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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 ≈ -0.88443731 and x ≈ 1.8844373.

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 ≈ -0.884437, 1.88444

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

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Graph of y = -3x² + 3x + 5-3-2-11234-40-30-20-1010x = -0.88443731x = 1.8844373vertexy = -3x² + 3x + 5
Graph of y = -3x² + 3x + 5. The curve crosses the x-axis at x = -0.88443731 and x = 1.8844373; 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=32−4⋅(−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 ≈ -0.88443731 and x ≈ 1.8844373.

  7. 7

    Check numerically: for x ≈ -0.88443731 the left side is 0 (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
-2.5-21.25
-1.5-6.25
-0.52.75
0.55.75
1.52.75
2.5-6.25
3.5-21.25

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 ≈ -0.88443731 and x ≈ 1.8844373.

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