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Solve x² - x - 2 = 0

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

x² +x += 0
x2−x−2=0⇒x=2, −1x^2 - x - 2 = 0 \Rightarrow x = 2,\ -1
Discriminant D = 9 · 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 - x - 2-3-2-11234-4-22468101214x = 2x = -1vertexy = x² - x - 2
Graph of y = x² - x - 2. The curve crosses the x-axis at x = 2 and x = -1; the vertex is at (0.5, -2.25).

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 = -1, c = -2.

    x2−x−2=0x^2 - x - 2 = 0
  2. 2

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

    D=(−1)2−4⋅1⋅(−2)=1+8=9D = (-1)^2 - 4\cdot1\cdot(-2) = 1 + 8 = 9
  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=1±92x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{9}}{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) + (-1)·(2) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - x - 2 = 0\ \Rightarrow\ x = 2,\ -1 \]
\[ x^2 - x - 2 = 0 \]
\[ D = (-1)^2 - 4\cdot1\cdot(-2) = 1 + 8 = 9 \]
\[ \left(x - 2\right)\left(x + 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{9}}{2} \]
\[ x = 2\quad\text{or}\quad x = -1 \]

The parabola y = x² - x - 2

xy
-2.56.75
-1.51.75
-0.5-1.25
0.5-2.25
1.5-1.25
2.51.75
3.56.75

Questions

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

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

Does x² - x - 2 = 0 factor?

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

What is the vertex of y = x² - x - 2?

(1/2, -9/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -2.

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