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Solve x² - 6x - 2 = 0

The solutions of x² - 6x - 2 = 0 are x = 3 + √11 or x = 3 - √11, that is x ≈ 6.3166248 and x ≈ -0.31662479.

x² +x += 0
x2−6x−2=0⇒x=3+11, 3−11x^2 - 6x - 2 = 0 \Rightarrow x = 3 + \sqrt{11},\ 3 - \sqrt{11}
Discriminant D = 44 · x ≈ 6.31662, -0.316625

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

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Graph of y = x² - 6x - 2-22468-15-10-5510152025x = 6.3166248x = -0.31662479vertexy = x² - 6x - 2
Graph of y = x² - 6x - 2. The curve crosses the x-axis at x = 6.3166248 and x = -0.31662479; the vertex is at (3, -11).

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 = -6, c = -2.

    x2−6x−2=0x^2 - 6x - 2 = 0
  2. 2

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

    D=(−6)2−4⋅1⋅(−2)=36+8=44D = (-6)^2 - 4\cdot1\cdot(-2) = 36 + 8 = 44
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=6±442x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=3+11orx=3−11x = 3 + \sqrt{11}\quad\text{or}\quad x = 3 - \sqrt{11}
  6. 6

    As decimals: x ≈ 6.3166248 and x ≈ -0.31662479.

  7. 7

    Check numerically: for x ≈ 6.3166248 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 6x - 2 = 0\ \Rightarrow\ x = 3 + \sqrt{11},\ 3 - \sqrt{11} \]
\[ x^2 - 6x - 2 = 0 \]
\[ D = (-6)^2 - 4\cdot1\cdot(-2) = 36 + 8 = 44 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{2} \]
\[ x = 3 + \sqrt{11}\quad\text{or}\quad x = 3 - \sqrt{11} \]

The parabola y = x² - 6x - 2

xy
0-2
1-7
2-10
3-11
4-10
5-7
6-2

Questions

What are the solutions of x² - 6x - 2 = 0?

The solutions of x² - 6x - 2 = 0 are x = 3 + √11 or x = 3 - √11, that is x ≈ 6.3166248 and x ≈ -0.31662479.

Does x² - 6x - 2 = 0 factor?

Not over the integers, because the discriminant 44 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 6x - 2?

(3, -11). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = -2.

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