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Solve 3x² + 11x + 8 = 0

The solutions of 3x² + 11x + 8 = 0 are x = -1 or x = -8/3.

x² +x += 0
3x2+11x+8=0⇒x=−1, −833x^2 + 11x + 8 = 0 \Rightarrow x = -1,\ -\frac{8}{3}
Discriminant D = 25 · x ≈ -1, -2.66667

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

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Graph of y = 3x² + 11x + 8-5-4-3-2-11-551015202530x = -1x = -2.6666667vertexy = 3x² + 11x + 8
Graph of y = 3x² + 11x + 8. The curve crosses the x-axis at x = -1 and x = -2.6666667; the vertex is at (-1.8333333, -2.0833333).

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=112−4⋅3⋅8=121−96=25D = 11^2 - 4\cdot3\cdot8 = 121 - 96 = 25
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x+1)(3x+8)=0\left(x + 1\right)\left(3x + 8\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+1=0⇒x=−13x+8=0⇒x=−83x + 1 = 0 \Rightarrow x = -1\qquad 3x + 8 = 0 \Rightarrow x = -\frac{8}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±256x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{25}}{6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−83x = -1\quad\text{or}\quad x = -\frac{8}{3}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 3·(1) + 11·(-1) + 8 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 11x + 8 = 0\ \Rightarrow\ x = -1,\ -\frac{8}{3} \]
\[ 3x^2 + 11x + 8 = 0 \]
\[ D = 11^2 - 4\cdot3\cdot8 = 121 - 96 = 25 \]
\[ \left(x + 1\right)\left(3x + 8\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x + 8 = 0 \Rightarrow x = -\frac{8}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{25}}{6} \]
\[ x = -1\quad\text{or}\quad x = -\frac{8}{3} \]

The parabola y = 3x² + 11x + 8

xy
-4.833324.916067
-3.83339.9162667
-2.83330.91646667
-1.8333-2.0833333
-0.83330.91686667
0.16679.9170667
1.166724.917267

Questions

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

The solutions of 3x² + 11x + 8 = 0 are x = -1 or x = -8/3.

Does 3x² + 11x + 8 = 0 factor?

Yes: (x+1)(3x+8) = 0.

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

(-11/6, -25/12). 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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