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Solve 3x² + 11x - 9 = 0

The solutions of 3x² + 11x - 9 = 0 are x = (-11 + √229)/6 or x = (-11 - √229)/6, that is x ≈ 0.68879099 and x ≈ -4.3554577.

x² +x += 0
3x2+11x−9=0⇒x=−11+2296, −11−22963x^2 + 11x - 9 = 0 \Rightarrow x = \frac{-11 + \sqrt{229}}{6},\ \frac{-11 - \sqrt{229}}{6}
Discriminant D = 229 · x ≈ 0.688791, -4.35546

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

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Graph of y = 3x² + 11x - 9-6-5-4-3-2-1123-20204060x = 0.68879099x = -4.3554577vertexy = 3x² + 11x - 9
Graph of y = 3x² + 11x - 9. The curve crosses the x-axis at x = 0.68879099 and x = -4.3554577; the vertex is at (-1.8333333, -19.083333).

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 = 11, c = -9.

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

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

    D=112−4⋅3⋅(−9)=121+108=229D = 11^2 - 4\cdot3\cdot(-9) = 121 + 108 = 229
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−11±2296x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{229}}{6}
  5. 5

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

    x=−11+2296orx=−11−2296x = \frac{-11 + \sqrt{229}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{229}}{6}
  6. 6

    As decimals: x ≈ 0.68879099 and x ≈ -4.3554577.

  7. 7

    Check numerically: for x ≈ 0.68879099 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 11x - 9 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{229}}{6},\ \frac{-11 - \sqrt{229}}{6} \]
\[ 3x^2 + 11x - 9 = 0 \]
\[ D = 11^2 - 4\cdot3\cdot(-9) = 121 + 108 = 229 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{229}}{6} \]
\[ x = \frac{-11 + \sqrt{229}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{229}}{6} \]

The parabola y = 3x² + 11x - 9

xy
-4.83337.9160667
-3.8333-7.0837333
-2.8333-16.083533
-1.8333-19.083333
-0.8333-16.083133
0.1667-7.0829333
1.16677.9172667

Questions

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

The solutions of 3x² + 11x - 9 = 0 are x = (-11 + √229)/6 or x = (-11 - √229)/6, that is x ≈ 0.68879099 and x ≈ -4.3554577.

Does 3x² + 11x - 9 = 0 factor?

Not over the integers, because the discriminant 229 is not a perfect square. Use the quadratic formula.

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

(-11/6, -229/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/3, product = c/a = -3.

Similar equations

    Solve 3x² + 11x - 9 = 0: x = (-11 + √229)/6 or x = (-11 - √229)/6 (steps) | WordGPT