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Solve 5x² + 11x + 6 = 0

The solutions of 5x² + 11x + 6 = 0 are x = -1 or x = -6/5.

x² +x += 0
5x2+11x+6=0⇒x=−1, −655x^2 + 11x + 6 = 0 \Rightarrow x = -1,\ -\frac{6}{5}
Discriminant D = 1 · x ≈ -1, -1.2

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

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Graph of y = 5x² + 11x + 6-3.5-3-2.5-2-1.5-1-0.50.511.55101520253035x = -1x = -1.2vertexy = 5x² + 11x + 6
Graph of y = 5x² + 11x + 6. The curve crosses the x-axis at x = -1 and x = -1.2; the vertex is at (-1.1, -0.05).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = 11, c = 6.

    5x2+11x+6=05x^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⋅5⋅6=121−120=1D = 11^2 - 4\cdot5\cdot6 = 121 - 120 = 1
  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)(5x+6)=0\left(x + 1\right)\left(5x + 6\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=−15x+6=0⇒x=−65x + 1 = 0 \Rightarrow x = -1\qquad 5x + 6 = 0 \Rightarrow x = -\frac{6}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{1}}{10}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−65x = -1\quad\text{or}\quad x = -\frac{6}{5}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 5·(1) + 11·(-1) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 11x + 6 = 0\ \Rightarrow\ x = -1,\ -\frac{6}{5} \]
\[ 5x^2 + 11x + 6 = 0 \]
\[ D = 11^2 - 4\cdot5\cdot6 = 121 - 120 = 1 \]
\[ \left(x + 1\right)\left(5x + 6\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 5x + 6 = 0 \Rightarrow x = -\frac{6}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{1}}{10} \]
\[ x = -1\quad\text{or}\quad x = -\frac{6}{5} \]

The parabola y = 5x² + 11x + 6

xy
-4.144.95
-3.119.95
-2.14.95
-1.1-0.05
-0.14.95
0.919.95
1.944.95

Questions

What are the solutions of 5x² + 11x + 6 = 0?

The solutions of 5x² + 11x + 6 = 0 are x = -1 or x = -6/5.

Does 5x² + 11x + 6 = 0 factor?

Yes: (x+1)(5x+6) = 0.

What is the vertex of y = 5x² + 11x + 6?

(-11/10, -1/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/5, product = c/a = 6/5.

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