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Solve -x² - 3x - 2 = 0

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

x² +x += 0
−x2−3x−2=0⇒x=−2, −1-x^2 - 3x - 2 = 0 \Rightarrow x = -2,\ -1
Discriminant D = 1 · x ≈ -2, -1

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

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Graph of y = -x² - 3x - 2-4-3-2-11-8-6-4-2x = -2x = -1vertexy = -x² - 3x - 2
Graph of y = -x² - 3x - 2. The curve crosses the x-axis at x = -2 and x = -1; the vertex is at (-1.5, 0.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -3, c = -2.

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

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

    D=(−3)2−4⋅(−1)⋅(−2)=9−8=1D = (-3)^2 - 4\cdot(-1)\cdot(-2) = 9 - 8 = 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+2)(x+1)=0-\left(x + 2\right)\left(x + 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+2=0⇒x=−2x+1=0⇒x=−1x + 2 = 0 \Rightarrow x = -2\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±1−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{1}}{-2}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -2 back in: a·x² + b·x + c = -1·(4) + (-3)·(-2) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 3x - 2 = 0\ \Rightarrow\ x = -2,\ -1 \]
\[ -x^2 - 3x - 2 = 0 \]
\[ D = (-3)^2 - 4\cdot(-1)\cdot(-2) = 9 - 8 = 1 \]
\[ -\left(x + 2\right)\left(x + 1\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{1}}{-2} \]
\[ x = -2\quad\text{or}\quad x = -1 \]

The parabola y = -x² - 3x - 2

xy
-4.5-8.75
-3.5-3.75
-2.5-0.75
-1.50.25
-0.5-0.75
0.5-3.75
1.5-8.75

Questions

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

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

Does -x² - 3x - 2 = 0 factor?

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

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

(-3/2, 1/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 2.

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