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Solve x² - 9x - 11 = 0

The solutions of x² - 9x - 11 = 0 are x = (9 + 5√5)/2 or x = (9 - 5√5)/2, that is x ≈ 10.09017 and x ≈ -1.0901699.

x² +x += 0
x2−9x−11=0⇒x=9+552, 9−552x^2 - 9x - 11 = 0 \Rightarrow x = \frac{9 + 5\sqrt{5}}{2},\ \frac{9 - 5\sqrt{5}}{2}
Discriminant D = 125 · x ≈ 10.0902, -1.09017

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

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Graph of y = x² - 9x - 11-224681012-30-20-10102030x = 10.09017x = -1.0901699vertexy = x² - 9x - 11
Graph of y = x² - 9x - 11. The curve crosses the x-axis at x = 10.09017 and x = -1.0901699; the vertex is at (4.5, -31.25).

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 = -9, c = -11.

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

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

    D=(−9)2−4⋅1⋅(−11)=81+44=125D = (-9)^2 - 4\cdot1\cdot(-11) = 81 + 44 = 125
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=9±1252x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{125}}{2}
  5. 5

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

    x=9+552orx=9−552x = \frac{9 + 5\sqrt{5}}{2}\quad\text{or}\quad x = \frac{9 - 5\sqrt{5}}{2}
  6. 6

    As decimals: x ≈ 10.09017 and x ≈ -1.0901699.

  7. 7

    Check numerically: for x ≈ 10.09017 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 9x - 11 = 0\ \Rightarrow\ x = \frac{9 + 5\sqrt{5}}{2},\ \frac{9 - 5\sqrt{5}}{2} \]
\[ x^2 - 9x - 11 = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot(-11) = 81 + 44 = 125 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{125}}{2} \]
\[ x = \frac{9 + 5\sqrt{5}}{2}\quad\text{or}\quad x = \frac{9 - 5\sqrt{5}}{2} \]

The parabola y = x² - 9x - 11

xy
1.5-22.25
2.5-27.25
3.5-30.25
4.5-31.25
5.5-30.25
6.5-27.25
7.5-22.25

Questions

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

The solutions of x² - 9x - 11 = 0 are x = (9 + 5√5)/2 or x = (9 - 5√5)/2, that is x ≈ 10.09017 and x ≈ -1.0901699.

Does x² - 9x - 11 = 0 factor?

Not over the integers, because the discriminant 125 is not a perfect square. Use the quadratic formula.

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

(9/2, -125/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9, product = c/a = -11.

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    Solve x² - 9x - 11 = 0: x = (9 + 5√5)/2 or x = (9 - 5√5)/2 (steps) | WordGPT