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Solve x² - 10x + 11 = 0

The solutions of x² - 10x + 11 = 0 are x = 5 + √14 or x = 5 - √14, that is x ≈ 8.7416574 and x ≈ 1.2583426.

x² +x += 0
x2−10x+11=0⇒x=5+14, 5−14x^2 - 10x + 11 = 0 \Rightarrow x = 5 + \sqrt{14},\ 5 - \sqrt{14}
Discriminant D = 56 · x ≈ 8.74166, 1.25834

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

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Graph of y = x² - 10x + 11246810-101020x = 8.7416574x = 1.2583426vertexy = x² - 10x + 11
Graph of y = x² - 10x + 11. The curve crosses the x-axis at x = 8.7416574 and x = 1.2583426; the vertex is at (5, -14).

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 = 11.

    x2−10x+11=0x^2 - 10x + 11 = 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⋅11=100−44=56D = (-10)^2 - 4\cdot1\cdot11 = 100 - 44 = 56
  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±562x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{56}}{2}
  5. 5

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

    x=5+14orx=5−14x = 5 + \sqrt{14}\quad\text{or}\quad x = 5 - \sqrt{14}
  6. 6

    As decimals: x ≈ 8.7416574 and x ≈ 1.2583426.

  7. 7

    Check numerically: for x ≈ 8.7416574 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x + 11 = 0\ \Rightarrow\ x = 5 + \sqrt{14},\ 5 - \sqrt{14} \]
\[ x^2 - 10x + 11 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot11 = 100 - 44 = 56 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{56}}{2} \]
\[ x = 5 + \sqrt{14}\quad\text{or}\quad x = 5 - \sqrt{14} \]

The parabola y = x² - 10x + 11

xy
2-5
3-10
4-13
5-14
6-13
7-10
8-5

Questions

What are the solutions of x² - 10x + 11 = 0?

The solutions of x² - 10x + 11 = 0 are x = 5 + √14 or x = 5 - √14, that is x ≈ 8.7416574 and x ≈ 1.2583426.

Does x² - 10x + 11 = 0 factor?

Not over the integers, because the discriminant 56 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x + 11?

(5, -14). 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 = 11.

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