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Solve x² - 8x - 2 = 0

The solutions of x² - 8x - 2 = 0 are x = 4 + 3√2 or x = 4 - 3√2, that is x ≈ 8.2426407 and x ≈ -0.24264069.

x² +x += 0
x2−8x−2=0⇒x=4+32, 4−32x^2 - 8x - 2 = 0 \Rightarrow x = 4 + 3\sqrt{2},\ 4 - 3\sqrt{2}
Discriminant D = 72 · x ≈ 8.24264, -0.242641

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

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Graph of y = x² - 8x - 2-2246810-20-10102030x = 8.2426407x = -0.24264069vertexy = x² - 8x - 2
Graph of y = x² - 8x - 2. The curve crosses the x-axis at x = 8.2426407 and x = -0.24264069; the vertex is at (4, -18).

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 = -8, c = -2.

    x2−8x−2=0x^2 - 8x - 2 = 0
  2. 2

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

    D=(−8)2−4⋅1⋅(−2)=64+8=72D = (-8)^2 - 4\cdot1\cdot(-2) = 64 + 8 = 72
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=8±722x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{72}}{2}
  5. 5

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

    x=4+32orx=4−32x = 4 + 3\sqrt{2}\quad\text{or}\quad x = 4 - 3\sqrt{2}
  6. 6

    As decimals: x ≈ 8.2426407 and x ≈ -0.24264069.

  7. 7

    Check numerically: for x ≈ 8.2426407 the left side is -1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x - 2 = 0\ \Rightarrow\ x = 4 + 3\sqrt{2},\ 4 - 3\sqrt{2} \]
\[ x^2 - 8x - 2 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot(-2) = 64 + 8 = 72 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{72}}{2} \]
\[ x = 4 + 3\sqrt{2}\quad\text{or}\quad x = 4 - 3\sqrt{2} \]

The parabola y = x² - 8x - 2

xy
1-9
2-14
3-17
4-18
5-17
6-14
7-9

Questions

What are the solutions of x² - 8x - 2 = 0?

The solutions of x² - 8x - 2 = 0 are x = 4 + 3√2 or x = 4 - 3√2, that is x ≈ 8.2426407 and x ≈ -0.24264069.

Does x² - 8x - 2 = 0 factor?

Not over the integers, because the discriminant 72 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 8x - 2?

(4, -18). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = -2.

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