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Solve 3x² - 11x - 8 = 0

The solutions of 3x² - 11x - 8 = 0 are x = (11 + √217)/6 or x = (11 - √217)/6, that is x ≈ 4.2884866 and x ≈ -0.62181998.

x² +x += 0
3x2−11x−8=0⇒x=11+2176, 11−21763x^2 - 11x - 8 = 0 \Rightarrow x = \frac{11 + \sqrt{217}}{6},\ \frac{11 - \sqrt{217}}{6}
Discriminant D = 217 · x ≈ 4.28849, -0.62182

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

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Graph of y = 3x² - 11x - 8-3-2-1123456-20-10102030405060x = 4.2884866x = -0.62181998vertexy = 3x² - 11x - 8
Graph of y = 3x² - 11x - 8. The curve crosses the x-axis at x = 4.2884866 and x = -0.62181998; the vertex is at (1.8333333, -18.083333).

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 = -8.

    3x2−11x−8=03x^2 - 11x - 8 = 0
  2. 2

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

    D=(−11)2−4⋅3⋅(−8)=121+96=217D = (-11)^2 - 4\cdot3\cdot(-8) = 121 + 96 = 217
  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±2176x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{217}}{6}
  5. 5

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

    x=11+2176orx=11−2176x = \frac{11 + \sqrt{217}}{6}\quad\text{or}\quad x = \frac{11 - \sqrt{217}}{6}
  6. 6

    As decimals: x ≈ 4.2884866 and x ≈ -0.62181998.

  7. 7

    Check numerically: for x ≈ 4.2884866 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 11x - 8 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{217}}{6},\ \frac{11 - \sqrt{217}}{6} \]
\[ 3x^2 - 11x - 8 = 0 \]
\[ D = (-11)^2 - 4\cdot3\cdot(-8) = 121 + 96 = 217 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{217}}{6} \]
\[ x = \frac{11 + \sqrt{217}}{6}\quad\text{or}\quad x = \frac{11 - \sqrt{217}}{6} \]

The parabola y = 3x² - 11x - 8

xy
-1.16678.9172667
-0.1667-6.0829333
0.8333-15.083133
1.8333-18.083333
2.8333-15.083533
3.8333-6.0837333
4.83338.9160667

Questions

What are the solutions of 3x² - 11x - 8 = 0?

The solutions of 3x² - 11x - 8 = 0 are x = (11 + √217)/6 or x = (11 - √217)/6, that is x ≈ 4.2884866 and x ≈ -0.62181998.

Does 3x² - 11x - 8 = 0 factor?

Not over the integers, because the discriminant 217 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 11x - 8?

(11/6, -217/12). 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 = -8/3.

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    Solve 3x² - 11x - 8 = 0: x = (11 + √217)/6 or x = (11 - √217)/6 (steps) | WordGPT