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Solve x² - 10x - 1 = 0

The solutions of x² - 10x - 1 = 0 are x = 5 + √26 or x = 5 - √26, that is x ≈ 10.09902 and x ≈ -0.099019514.

x² +x += 0
x2−10x−1=0⇒x=5+26, 5−26x^2 - 10x - 1 = 0 \Rightarrow x = 5 + \sqrt{26},\ 5 - \sqrt{26}
Discriminant D = 104 · x ≈ 10.099, -0.0990195

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

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Graph of y = x² - 10x - 1-224681012-30-20-10102030x = 10.09902x = -0.099019514vertexy = x² - 10x - 1
Graph of y = x² - 10x - 1. The curve crosses the x-axis at x = 10.09902 and x = -0.099019514; the vertex is at (5, -26).

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 = -10, c = -1.

    x2−10x−1=0x^2 - 10x - 1 = 0
  2. 2

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

    D=(−10)2−4⋅1⋅(−1)=100+4=104D = (-10)^2 - 4\cdot1\cdot(-1) = 100 + 4 = 104
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=10±1042x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{104}}{2}
  5. 5

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

    x=5+26orx=5−26x = 5 + \sqrt{26}\quad\text{or}\quad x = 5 - \sqrt{26}
  6. 6

    As decimals: x ≈ 10.09902 and x ≈ -0.099019514.

  7. 7

    Check numerically: for x ≈ 10.09902 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x - 1 = 0\ \Rightarrow\ x = 5 + \sqrt{26},\ 5 - \sqrt{26} \]
\[ x^2 - 10x - 1 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot(-1) = 100 + 4 = 104 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{104}}{2} \]
\[ x = 5 + \sqrt{26}\quad\text{or}\quad x = 5 - \sqrt{26} \]

The parabola y = x² - 10x - 1

xy
2-17
3-22
4-25
5-26
6-25
7-22
8-17

Questions

What are the solutions of x² - 10x - 1 = 0?

The solutions of x² - 10x - 1 = 0 are x = 5 + √26 or x = 5 - √26, that is x ≈ 10.09902 and x ≈ -0.099019514.

Does x² - 10x - 1 = 0 factor?

Not over the integers, because the discriminant 104 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x - 1?

(5, -26). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 10, product = c/a = -1.

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