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Solve 3x² + 9x - 11 = 0

The solutions of 3x² + 9x - 11 = 0 are x = (-9 + √213)/6 or x = (-9 - √213)/6, that is x ≈ 0.93241992 and x ≈ -3.9324199.

x² +x += 0
3x2+9x−11=0⇒x=−9+2136, −9−21363x^2 + 9x - 11 = 0 \Rightarrow x = \frac{-9 + \sqrt{213}}{6},\ \frac{-9 - \sqrt{213}}{6}
Discriminant D = 213 · x ≈ 0.93242, -3.93242

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

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Graph of y = 3x² + 9x - 11-6-5-4-3-2-1123-20-10102030405060x = 0.93241992x = -3.9324199vertexy = 3x² + 9x - 11
Graph of y = 3x² + 9x - 11. The curve crosses the x-axis at x = 0.93241992 and x = -3.9324199; the vertex is at (-1.5, -17.75).

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 = -11.

    3x2+9x−11=03x^2 + 9x - 11 = 0
  2. 2

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

    D=92−4⋅3⋅(−11)=81+132=213D = 9^2 - 4\cdot3\cdot(-11) = 81 + 132 = 213
  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±2136x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{213}}{6}
  5. 5

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

    x=−9+2136orx=−9−2136x = \frac{-9 + \sqrt{213}}{6}\quad\text{or}\quad x = \frac{-9 - \sqrt{213}}{6}
  6. 6

    As decimals: x ≈ 0.93241992 and x ≈ -3.9324199.

  7. 7

    Check numerically: for x ≈ 0.93241992 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 9x - 11 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{213}}{6},\ \frac{-9 - \sqrt{213}}{6} \]
\[ 3x^2 + 9x - 11 = 0 \]
\[ D = 9^2 - 4\cdot3\cdot(-11) = 81 + 132 = 213 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{213}}{6} \]
\[ x = \frac{-9 + \sqrt{213}}{6}\quad\text{or}\quad x = \frac{-9 - \sqrt{213}}{6} \]

The parabola y = 3x² + 9x - 11

xy
-4.59.25
-3.5-5.75
-2.5-14.75
-1.5-17.75
-0.5-14.75
0.5-5.75
1.59.25

Questions

What are the solutions of 3x² + 9x - 11 = 0?

The solutions of 3x² + 9x - 11 = 0 are x = (-9 + √213)/6 or x = (-9 - √213)/6, that is x ≈ 0.93241992 and x ≈ -3.9324199.

Does 3x² + 9x - 11 = 0 factor?

Not over the integers, because the discriminant 213 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 9x - 11?

(-3/2, -71/4). 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 = -11/3.

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    Solve 3x² + 9x - 11 = 0: x = (-9 + √213)/6 or x = (-9 - √213)/6 (steps) | WordGPT