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Solve -3x² - 5x - 2 = 0

The solutions of -3x² - 5x - 2 = 0 are x = -1 or x = -2/3.

x² +x += 0
−3x2−5x−2=0⇒x=−1, −23-3x^2 - 5x - 2 = 0 \Rightarrow x = -1,\ -\frac{2}{3}
Discriminant D = 1 · x ≈ -1, -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 - 5x - 2-3-2-11-20-15-10-5x = -1x = -0.66666667vertexy = -3x² - 5x - 2
Graph of y = -3x² - 5x - 2. The curve crosses the x-axis at x = -1 and x = -0.66666667; the vertex is at (-0.83333333, 0.083333333).

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 = -5, c = -2.

    −3x2−5x−2=0-3x^2 - 5x - 2 = 0
  2. 2

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

    D=(−5)2−4⋅(−3)⋅(−2)=25−24=1D = (-5)^2 - 4\cdot(-3)\cdot(-2) = 25 - 24 = 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)(3x+2)=0-\left(x + 1\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+1=0⇒x=−13x+2=0⇒x=−23x + 1 = 0 \Rightarrow x = -1\qquad 3x + 2 = 0 \Rightarrow x = -\frac{2}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±1−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−23x = -1\quad\text{or}\quad x = -\frac{2}{3}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -3·(1) + (-5)·(-1) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 5x - 2 = 0\ \Rightarrow\ x = -1,\ -\frac{2}{3} \]
\[ -3x^2 - 5x - 2 = 0 \]
\[ D = (-5)^2 - 4\cdot(-3)\cdot(-2) = 25 - 24 = 1 \]
\[ -\left(x + 1\right)\left(3x + 2\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x + 2 = 0 \Rightarrow x = -\frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{-6} \]
\[ x = -1\quad\text{or}\quad x = -\frac{2}{3} \]

The parabola y = -3x² - 5x - 2

xy
-3.8333-26.916067
-2.8333-11.916267
-1.8333-2.9164667
-0.83330.08333333
0.1667-2.9168667
1.1667-11.917067
2.1667-26.917267

Questions

What are the solutions of -3x² - 5x - 2 = 0?

The solutions of -3x² - 5x - 2 = 0 are x = -1 or x = -2/3.

Does -3x² - 5x - 2 = 0 factor?

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

What is the vertex of y = -3x² - 5x - 2?

(-5/6, 1/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5/3, product = c/a = 2/3.

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