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Solve -3x² + 3x + 9 = 0

The solutions of -3x² + 3x + 9 = 0 are x = (1 + √13)/2 or x = (1 - √13)/2, that is x ≈ -1.3027756 and x ≈ 2.3027756.

x² +x += 0
−3x2+3x+9=0⇒x=1+132, 1−132-3x^2 + 3x + 9 = 0 \Rightarrow x = \frac{1 + \sqrt{13}}{2},\ \frac{1 - \sqrt{13}}{2}
Discriminant D = 117 · x ≈ -1.30278, 2.30278

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

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Graph of y = -3x² + 3x + 9-3-2-11234-40-30-20-1010x = -1.3027756x = 2.3027756vertexy = -3x² + 3x + 9
Graph of y = -3x² + 3x + 9. The curve crosses the x-axis at x = -1.3027756 and x = 2.3027756; the vertex is at (0.5, 9.75).

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 = 9.

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

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

    D=32−4⋅(−3)⋅9=9+108=117D = 3^2 - 4\cdot(-3)\cdot9 = 9 + 108 = 117
  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±117−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{117}}{-6}
  5. 5

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

    x=1+132orx=1−132x = \frac{1 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{13}}{2}
  6. 6

    As decimals: x ≈ -1.3027756 and x ≈ 2.3027756.

  7. 7

    Check numerically: for x ≈ -1.3027756 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 3x + 9 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{13}}{2},\ \frac{1 - \sqrt{13}}{2} \]
\[ -3x^2 + 3x + 9 = 0 \]
\[ D = 3^2 - 4\cdot(-3)\cdot9 = 9 + 108 = 117 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{117}}{-6} \]
\[ x = \frac{1 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{13}}{2} \]

The parabola y = -3x² + 3x + 9

xy
-2.5-17.25
-1.5-2.25
-0.56.75
0.59.75
1.56.75
2.5-2.25
3.5-17.25

Questions

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

The solutions of -3x² + 3x + 9 = 0 are x = (1 + √13)/2 or x = (1 - √13)/2, that is x ≈ -1.3027756 and x ≈ 2.3027756.

Does -3x² + 3x + 9 = 0 factor?

Not over the integers, because the discriminant 117 is not a perfect square. Use the quadratic formula.

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

(1/2, 39/4). 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 = -3.

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