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Solve -x² - 2x - 1 = 0

The solution of -x² - 2x - 1 = 0 is x = -1 (a repeated root).

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

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

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Graph of y = -x² - 2x - 1-3.5-3-2.5-2-1.5-1-0.50.511.5-6-5-4-3-2-1x = -1vertexy = -x² - 2x - 1
Graph of y = -x² - 2x - 1. The curve crosses the x-axis at x = -1; the vertex is at (-1, 0).

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 = -2, c = -1.

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

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

    D=(−2)2−4⋅(−1)⋅(−1)=4−4=0D = (-2)^2 - 4\cdot(-1)\cdot(-1) = 4 - 4 = 0
  3. 3

    D is zero, so there is exactly one solution (a repeated root).

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x+1)(x+1)=0-\left(x + 1\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+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=2±0−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{0}}{-2}
  7. 7

    Simplify to get the solution:

    x=−1x = -1
  8. 8

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

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

The parabola y = -x² - 2x - 1

xy
-4-9
-3-4
-2-1
-10
0-1
1-4
2-9

Questions

What are the solutions of -x² - 2x - 1 = 0?

The solution of -x² - 2x - 1 = 0 is x = -1 (a repeated root).

Does -x² - 2x - 1 = 0 factor?

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

What is the vertex of y = -x² - 2x - 1?

(-1, 0). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = 1.

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