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Solve x² - 8x - 10 = 0

The solutions of x² - 8x - 10 = 0 are x = 4 + √26 or x = 4 - √26, that is x ≈ 9.0990195 and x ≈ -1.0990195.

x² +x += 0
x2−8x−10=0⇒x=4+26, 4−26x^2 - 8x - 10 = 0 \Rightarrow x = 4 + \sqrt{26},\ 4 - \sqrt{26}
Discriminant D = 104 · x ≈ 9.09902, -1.09902

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

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Graph of y = x² - 8x - 10-2246810-30-20-10102030x = 9.0990195x = -1.0990195vertexy = x² - 8x - 10
Graph of y = x² - 8x - 10. The curve crosses the x-axis at x = 9.0990195 and x = -1.0990195; the vertex is at (4, -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 = -8, c = -10.

    x2−8x−10=0x^2 - 8x - 10 = 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⋅(−10)=64+40=104D = (-8)^2 - 4\cdot1\cdot(-10) = 64 + 40 = 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=8±1042x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{104}}{2}
  5. 5

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

    x=4+26orx=4−26x = 4 + \sqrt{26}\quad\text{or}\quad x = 4 - \sqrt{26}
  6. 6

    As decimals: x ≈ 9.0990195 and x ≈ -1.0990195.

  7. 7

    Check numerically: for x ≈ 9.0990195 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x - 10 = 0\ \Rightarrow\ x = 4 + \sqrt{26},\ 4 - \sqrt{26} \]
\[ x^2 - 8x - 10 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot(-10) = 64 + 40 = 104 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{104}}{2} \]
\[ x = 4 + \sqrt{26}\quad\text{or}\quad x = 4 - \sqrt{26} \]

The parabola y = x² - 8x - 10

xy
1-17
2-22
3-25
4-26
5-25
6-22
7-17

Questions

What are the solutions of x² - 8x - 10 = 0?

The solutions of x² - 8x - 10 = 0 are x = 4 + √26 or x = 4 - √26, that is x ≈ 9.0990195 and x ≈ -1.0990195.

Does x² - 8x - 10 = 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 - 8x - 10?

(4, -26). 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 = -10.

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