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Solve 3x² - 9x - 9 = 0

The solutions of 3x² - 9x - 9 = 0 are x = (3 + √21)/2 or x = (3 - √21)/2, that is x ≈ 3.7912878 and x ≈ -0.79128785.

x² +x += 0
3x2−9x−9=0⇒x=3+212, 3−2123x^2 - 9x - 9 = 0 \Rightarrow x = \frac{3 + \sqrt{21}}{2},\ \frac{3 - \sqrt{21}}{2}
Discriminant D = 189 · x ≈ 3.79129, -0.791288

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

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Graph of y = 3x² - 9x - 9-3-2-1123456-20-101020304050x = 3.7912878x = -0.79128785vertexy = 3x² - 9x - 9
Graph of y = 3x² - 9x - 9. The curve crosses the x-axis at x = 3.7912878 and x = -0.79128785; the vertex is at (1.5, -15.75).

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 = -9.

    3x2−9x−9=03x^2 - 9x - 9 = 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⋅(−9)=81+108=189D = (-9)^2 - 4\cdot3\cdot(-9) = 81 + 108 = 189
  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±1896x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{189}}{6}
  5. 5

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

    x=3+212orx=3−212x = \frac{3 + \sqrt{21}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{21}}{2}
  6. 6

    As decimals: x ≈ 3.7912878 and x ≈ -0.79128785.

  7. 7

    Check numerically: for x ≈ 3.7912878 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 9x - 9 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{21}}{2},\ \frac{3 - \sqrt{21}}{2} \]
\[ 3x^2 - 9x - 9 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot(-9) = 81 + 108 = 189 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{189}}{6} \]
\[ x = \frac{3 + \sqrt{21}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{21}}{2} \]

The parabola y = 3x² - 9x - 9

xy
-1.511.25
-0.5-3.75
0.5-12.75
1.5-15.75
2.5-12.75
3.5-3.75
4.511.25

Questions

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

The solutions of 3x² - 9x - 9 = 0 are x = (3 + √21)/2 or x = (3 - √21)/2, that is x ≈ 3.7912878 and x ≈ -0.79128785.

Does 3x² - 9x - 9 = 0 factor?

Not over the integers, because the discriminant 189 is not a perfect square. Use the quadratic formula.

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

(3/2, -63/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -3.

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