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Solve 3x² + 11x + 6 = 0

The solutions of 3x² + 11x + 6 = 0 are x = -2/3 or x = -3.

x² +x += 0
3x2+11x+6=0⇒x=−23, −33x^2 + 11x + 6 = 0 \Rightarrow x = -\frac{2}{3},\ -3
Discriminant D = 49 · x ≈ -0.666667, -3

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

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Graph of y = 3x² + 11x + 6-5-4-3-2-11102030x = -0.66666667x = -3vertexy = 3x² + 11x + 6
Graph of y = 3x² + 11x + 6. The curve crosses the x-axis at x = -0.66666667 and x = -3; the vertex is at (-1.8333333, -4.0833333).

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 = 6.

    3x2+11x+6=03x^2 + 11x + 6 = 0
  2. 2

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

    D=112−4⋅3⋅6=121−72=49D = 11^2 - 4\cdot3\cdot6 = 121 - 72 = 49
  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:

    (3x+2)(x+3)=0\left(3x + 2\right)\left(x + 3\right) = 0
  5. 5

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

    3x+2=0⇒x=−23x+3=0⇒x=−33x + 2 = 0 \Rightarrow x = -\frac{2}{3}\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±496x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{6}
  7. 7

    Simplify to get the solutions:

    x=−23orx=−3x = -\frac{2}{3}\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = -2/3 back in: a·x² + b·x + c = 3·(4/9) + 11·(-2/3) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 11x + 6 = 0\ \Rightarrow\ x = -\frac{2}{3},\ -3 \]
\[ 3x^2 + 11x + 6 = 0 \]
\[ D = 11^2 - 4\cdot3\cdot6 = 121 - 72 = 49 \]
\[ \left(3x + 2\right)\left(x + 3\right) = 0 \]
\[ 3x + 2 = 0 \Rightarrow x = -\frac{2}{3}\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{6} \]
\[ x = -\frac{2}{3}\quad\text{or}\quad x = -3 \]

The parabola y = 3x² + 11x + 6

xy
-4.833322.916067
-3.83337.9162667
-2.8333-1.0835333
-1.8333-4.0833333
-0.8333-1.0831333
0.16677.9170667
1.166722.917267

Questions

What are the solutions of 3x² + 11x + 6 = 0?

The solutions of 3x² + 11x + 6 = 0 are x = -2/3 or x = -3.

Does 3x² + 11x + 6 = 0 factor?

Yes: (3x+2)(x+3) = 0.

What is the vertex of y = 3x² + 11x + 6?

(-11/6, -49/12). 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 = 2.

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