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Solve x² - 6x - 8 = 0

The solutions of x² - 6x - 8 = 0 are x = 3 + √17 or x = 3 - √17, that is x ≈ 7.1231056 and x ≈ -1.1231056.

x² +x += 0
x2−6x−8=0⇒x=3+17, 3−17x^2 - 6x - 8 = 0 \Rightarrow x = 3 + \sqrt{17},\ 3 - \sqrt{17}
Discriminant D = 68 · x ≈ 7.12311, -1.12311

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

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Graph of y = x² - 6x - 8-22468-20-10102030x = 7.1231056x = -1.1231056vertexy = x² - 6x - 8
Graph of y = x² - 6x - 8. The curve crosses the x-axis at x = 7.1231056 and x = -1.1231056; the vertex is at (3, -17).

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 = -8.

    x2−6x−8=0x^2 - 6x - 8 = 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⋅(−8)=36+32=68D = (-6)^2 - 4\cdot1\cdot(-8) = 36 + 32 = 68
  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±682x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{68}}{2}
  5. 5

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

    x=3+17orx=3−17x = 3 + \sqrt{17}\quad\text{or}\quad x = 3 - \sqrt{17}
  6. 6

    As decimals: x ≈ 7.1231056 and x ≈ -1.1231056.

  7. 7

    Check numerically: for x ≈ 7.1231056 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 6x - 8 = 0\ \Rightarrow\ x = 3 + \sqrt{17},\ 3 - \sqrt{17} \]
\[ x^2 - 6x - 8 = 0 \]
\[ D = (-6)^2 - 4\cdot1\cdot(-8) = 36 + 32 = 68 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{68}}{2} \]
\[ x = 3 + \sqrt{17}\quad\text{or}\quad x = 3 - \sqrt{17} \]

The parabola y = x² - 6x - 8

xy
0-8
1-13
2-16
3-17
4-16
5-13
6-8

Questions

What are the solutions of x² - 6x - 8 = 0?

The solutions of x² - 6x - 8 = 0 are x = 3 + √17 or x = 3 - √17, that is x ≈ 7.1231056 and x ≈ -1.1231056.

Does x² - 6x - 8 = 0 factor?

Not over the integers, because the discriminant 68 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 6x - 8?

(3, -17). 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 = -8.

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