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Solve 3x² + 10x - 11 = 0

The solutions of 3x² + 10x - 11 = 0 are x = (-5 + √58)/3 or x = (-5 - √58)/3, that is x ≈ 0.87192437 and x ≈ -4.2052577.

x² +x += 0
3x2+10x−11=0⇒x=−5+583, −5−5833x^2 + 10x - 11 = 0 \Rightarrow x = \frac{-5 + \sqrt{58}}{3},\ \frac{-5 - \sqrt{58}}{3}
Discriminant D = 232 · x ≈ 0.871924, -4.20526

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

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Graph of y = 3x² + 10x - 11-6-5-4-3-2-1123-20204060x = 0.87192437x = -4.2052577vertexy = 3x² + 10x - 11
Graph of y = 3x² + 10x - 11. The curve crosses the x-axis at x = 0.87192437 and x = -4.2052577; the vertex is at (-1.6666667, -19.333333).

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 = 10, c = -11.

    3x2+10x−11=03x^2 + 10x - 11 = 0
  2. 2

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

    D=102−4⋅3⋅(−11)=100+132=232D = 10^2 - 4\cdot3\cdot(-11) = 100 + 132 = 232
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±2326x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{232}}{6}
  5. 5

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

    x=−5+583orx=−5−583x = \frac{-5 + \sqrt{58}}{3}\quad\text{or}\quad x = \frac{-5 - \sqrt{58}}{3}
  6. 6

    As decimals: x ≈ 0.87192437 and x ≈ -4.2052577.

  7. 7

    Check numerically: for x ≈ 0.87192437 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 10x - 11 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{58}}{3},\ \frac{-5 - \sqrt{58}}{3} \]
\[ 3x^2 + 10x - 11 = 0 \]
\[ D = 10^2 - 4\cdot3\cdot(-11) = 100 + 132 = 232 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{232}}{6} \]
\[ x = \frac{-5 + \sqrt{58}}{3}\quad\text{or}\quad x = \frac{-5 - \sqrt{58}}{3} \]

The parabola y = 3x² + 10x - 11

xy
-4.66677.6672667
-3.6667-7.3329333
-2.6667-16.333133
-1.6667-19.333333
-0.6667-16.333533
0.3333-7.3337333
1.33337.6660667

Questions

What are the solutions of 3x² + 10x - 11 = 0?

The solutions of 3x² + 10x - 11 = 0 are x = (-5 + √58)/3 or x = (-5 - √58)/3, that is x ≈ 0.87192437 and x ≈ -4.2052577.

Does 3x² + 10x - 11 = 0 factor?

Not over the integers, because the discriminant 232 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 10x - 11?

(-5/3, -58/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -10/3, product = c/a = -11/3.

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    Solve 3x² + 10x - 11 = 0: x = (-5 + √58)/3 or x = (-5 - √58)/3 (steps) | WordGPT