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Solve 3x² + 11x - 11 = 0

The solutions of 3x² + 11x - 11 = 0 are x = (-11 + √253)/6 or x = (-11 - √253)/6, that is x ≈ 0.81766229 and x ≈ -4.484329.

x² +x += 0
3x2+11x−11=0⇒x=−11+2536, −11−25363x^2 + 11x - 11 = 0 \Rightarrow x = \frac{-11 + \sqrt{253}}{6},\ \frac{-11 - \sqrt{253}}{6}
Discriminant D = 253 · x ≈ 0.817662, -4.48433

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

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Graph of y = 3x² + 11x - 11-6-5-4-3-2-1123-20204060x = 0.81766229x = -4.484329vertexy = 3x² + 11x - 11
Graph of y = 3x² + 11x - 11. The curve crosses the x-axis at x = 0.81766229 and x = -4.484329; the vertex is at (-1.8333333, -21.083333).

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 = -11.

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

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

    D=112−4⋅3⋅(−11)=121+132=253D = 11^2 - 4\cdot3\cdot(-11) = 121 + 132 = 253
  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±2536x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{253}}{6}
  5. 5

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

    x=−11+2536orx=−11−2536x = \frac{-11 + \sqrt{253}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{253}}{6}
  6. 6

    As decimals: x ≈ 0.81766229 and x ≈ -4.484329.

  7. 7

    Check numerically: for x ≈ 0.81766229 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 11x - 11 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{253}}{6},\ \frac{-11 - \sqrt{253}}{6} \]
\[ 3x^2 + 11x - 11 = 0 \]
\[ D = 11^2 - 4\cdot3\cdot(-11) = 121 + 132 = 253 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{253}}{6} \]
\[ x = \frac{-11 + \sqrt{253}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{253}}{6} \]

The parabola y = 3x² + 11x - 11

xy
-4.83335.9160667
-3.8333-9.0837333
-2.8333-18.083533
-1.8333-21.083333
-0.8333-18.083133
0.1667-9.0829333
1.16675.9172667

Questions

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

The solutions of 3x² + 11x - 11 = 0 are x = (-11 + √253)/6 or x = (-11 - √253)/6, that is x ≈ 0.81766229 and x ≈ -4.484329.

Does 3x² + 11x - 11 = 0 factor?

Not over the integers, because the discriminant 253 is not a perfect square. Use the quadratic formula.

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

(-11/6, -253/12). 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 = -11/3.

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    Solve 3x² + 11x - 11 = 0: x = (-11 + √253)/6 or x = (-11 - √253)/6 (steps) | WordGPT