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Solve -3x² - 11x - 6 = 0

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

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

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

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Graph of y = -3x² - 11x - 6-5-4-3-2-11-30-20-10x = -3x = -0.66666667vertexy = -3x² - 11x - 6
Graph of y = -3x² - 11x - 6. The curve crosses the x-axis at x = -3 and x = -0.66666667; 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=0-3x^2 - 11x - 6 = 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)⋅(−6)=121−72=49D = (-11)^2 - 4\cdot(-3)\cdot(-6) = 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:

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

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

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

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

The parabola y = -3x² - 11x - 6

xy
-4.8333-22.916067
-3.8333-7.9162667
-2.83331.0835333
-1.83334.0833333
-0.83331.0831333
0.1667-7.9170667
1.1667-22.917267

Questions

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

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

Does -3x² - 11x - 6 = 0 factor?

Yes: -1(x+3)(3x+2) = 0.

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

(-11/6, 49/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -11/3, product = c/a = 2.

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