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Solve x² - x - 6 = 0

The solutions of x² - x - 6 = 0 are x = 3 or x = -2.

x² +x += 0
x2−x−6=0⇒x=3, −2x^2 - x - 6 = 0 \Rightarrow x = 3,\ -2
Discriminant D = 25 · x ≈ 3, -2

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

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Graph of y = x² - x - 6-4-3-2-112345-55101520x = 3x = -2vertexy = x² - x - 6
Graph of y = x² - x - 6. The curve crosses the x-axis at x = 3 and x = -2; the vertex is at (0.5, -6.25).

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 = -6.

    x2−x−6=0x^2 - x - 6 = 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⋅(−6)=1+24=25D = (-1)^2 - 4\cdot1\cdot(-6) = 1 + 24 = 25
  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−3)(x+2)=0\left(x - 3\right)\left(x + 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−3=0⇒x=3x+2=0⇒x=−2x - 3 = 0 \Rightarrow x = 3\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±252x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{2}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 3 back in: a·x² + b·x + c = 1·(9) + (-1)·(3) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - x - 6 = 0\ \Rightarrow\ x = 3,\ -2 \]
\[ x^2 - x - 6 = 0 \]
\[ D = (-1)^2 - 4\cdot1\cdot(-6) = 1 + 24 = 25 \]
\[ \left(x - 3\right)\left(x + 2\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{2} \]
\[ x = 3\quad\text{or}\quad x = -2 \]

The parabola y = x² - x - 6

xy
-2.52.75
-1.5-2.25
-0.5-5.25
0.5-6.25
1.5-5.25
2.5-2.25
3.52.75

Questions

What are the solutions of x² - x - 6 = 0?

The solutions of x² - x - 6 = 0 are x = 3 or x = -2.

Does x² - x - 6 = 0 factor?

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

What is the vertex of y = x² - x - 6?

(1/2, -25/4). 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 = -6.

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