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Solve -3x² - 4x - 1 = 0

The solutions of -3x² - 4x - 1 = 0 are x = -1 or x = -1/3.

x² +x += 0
−3x2−4x−1=0⇒x=−1, −13-3x^2 - 4x - 1 = 0 \Rightarrow x = -1,\ -\frac{1}{3}
Discriminant D = 4 · x ≈ -1, -0.333333

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

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Graph of y = -3x² - 4x - 1-3-2-112-25-20-15-10-5x = -1x = -0.33333333vertexy = -3x² - 4x - 1
Graph of y = -3x² - 4x - 1. The curve crosses the x-axis at x = -1 and x = -0.33333333; the vertex is at (-0.66666667, 0.33333333).

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 = -4, c = -1.

    −3x2−4x−1=0-3x^2 - 4x - 1 = 0
  2. 2

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

    D=(−4)2−4⋅(−3)⋅(−1)=16−12=4D = (-4)^2 - 4\cdot(-3)\cdot(-1) = 16 - 12 = 4
  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+1)=0-\left(x + 1\right)\left(3x + 1\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+1=0⇒x=−13x + 1 = 0 \Rightarrow x = -1\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±4−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{4}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−13x = -1\quad\text{or}\quad x = -\frac{1}{3}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -3·(1) + (-4)·(-1) + (-1) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 4x - 1 = 0\ \Rightarrow\ x = -1,\ -\frac{1}{3} \]
\[ -3x^2 - 4x - 1 = 0 \]
\[ D = (-4)^2 - 4\cdot(-3)\cdot(-1) = 16 - 12 = 4 \]
\[ -\left(x + 1\right)\left(3x + 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{4}}{-6} \]
\[ x = -1\quad\text{or}\quad x = -\frac{1}{3} \]

The parabola y = -3x² - 4x - 1

xy
-3.6667-26.667267
-2.6667-11.667067
-1.6667-2.6668667
-0.66670.33333333
0.3333-2.6664667
1.3333-11.666267
2.3333-26.666067

Questions

What are the solutions of -3x² - 4x - 1 = 0?

The solutions of -3x² - 4x - 1 = 0 are x = -1 or x = -1/3.

Does -3x² - 4x - 1 = 0 factor?

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

What is the vertex of y = -3x² - 4x - 1?

(-2/3, 1/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -4/3, product = c/a = 1/3.

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