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Solve x² - 11x + 9 = 0

The solutions of x² - 11x + 9 = 0 are x = (11 + √85)/2 or x = (11 - √85)/2, that is x ≈ 10.109772 and x ≈ 0.89022777.

x² +x += 0
x2−11x+9=0⇒x=11+852, 11−852x^2 - 11x + 9 = 0 \Rightarrow x = \frac{11 + \sqrt{85}}{2},\ \frac{11 - \sqrt{85}}{2}
Discriminant D = 85 · x ≈ 10.1098, 0.890228

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

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Graph of y = x² - 11x + 924681012-20-10102030x = 10.109772x = 0.89022777vertexy = x² - 11x + 9
Graph of y = x² - 11x + 9. The curve crosses the x-axis at x = 10.109772 and x = 0.89022777; the vertex is at (5.5, -21.25).

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 = -11, c = 9.

    x2−11x+9=0x^2 - 11x + 9 = 0
  2. 2

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

    D=(−11)2−4⋅1⋅9=121−36=85D = (-11)^2 - 4\cdot1\cdot9 = 121 - 36 = 85
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=11±852x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{85}}{2}
  5. 5

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

    x=11+852orx=11−852x = \frac{11 + \sqrt{85}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{85}}{2}
  6. 6

    As decimals: x ≈ 10.109772 and x ≈ 0.89022777.

  7. 7

    Check numerically: for x ≈ 10.109772 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 11x + 9 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{85}}{2},\ \frac{11 - \sqrt{85}}{2} \]
\[ x^2 - 11x + 9 = 0 \]
\[ D = (-11)^2 - 4\cdot1\cdot9 = 121 - 36 = 85 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{85}}{2} \]
\[ x = \frac{11 + \sqrt{85}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{85}}{2} \]

The parabola y = x² - 11x + 9

xy
2.5-12.25
3.5-17.25
4.5-20.25
5.5-21.25
6.5-20.25
7.5-17.25
8.5-12.25

Questions

What are the solutions of x² - 11x + 9 = 0?

The solutions of x² - 11x + 9 = 0 are x = (11 + √85)/2 or x = (11 - √85)/2, that is x ≈ 10.109772 and x ≈ 0.89022777.

Does x² - 11x + 9 = 0 factor?

Not over the integers, because the discriminant 85 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 11x + 9?

(11/2, -85/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11, product = c/a = 9.

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