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Solve x² - 12x - 11 = 0

The solutions of x² - 12x - 11 = 0 are x = 6 + √47 or x = 6 - √47, that is x ≈ 12.855655 and x ≈ -0.8556546.

x² +x += 0
x2−12x−11=0⇒x=6+47, 6−47x^2 - 12x - 11 = 0 \Rightarrow x = 6 + \sqrt{47},\ 6 - \sqrt{47}
Discriminant D = 188 · x ≈ 12.8557, -0.855655

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

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Graph of y = x² - 12x - 11-22468101214-40-202040x = 12.855655x = -0.8556546vertexy = x² - 12x - 11
Graph of y = x² - 12x - 11. The curve crosses the x-axis at x = 12.855655 and x = -0.8556546; the vertex is at (6, -47).

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 = -12, c = -11.

    x2−12x−11=0x^2 - 12x - 11 = 0
  2. 2

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

    D=(−12)2−4⋅1⋅(−11)=144+44=188D = (-12)^2 - 4\cdot1\cdot(-11) = 144 + 44 = 188
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±1882x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{188}}{2}
  5. 5

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

    x=6+47orx=6−47x = 6 + \sqrt{47}\quad\text{or}\quad x = 6 - \sqrt{47}
  6. 6

    As decimals: x ≈ 12.855655 and x ≈ -0.8556546.

  7. 7

    Check numerically: for x ≈ 12.855655 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 12x - 11 = 0\ \Rightarrow\ x = 6 + \sqrt{47},\ 6 - \sqrt{47} \]
\[ x^2 - 12x - 11 = 0 \]
\[ D = (-12)^2 - 4\cdot1\cdot(-11) = 144 + 44 = 188 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{188}}{2} \]
\[ x = 6 + \sqrt{47}\quad\text{or}\quad x = 6 - \sqrt{47} \]

The parabola y = x² - 12x - 11

xy
3-38
4-43
5-46
6-47
7-46
8-43
9-38

Questions

What are the solutions of x² - 12x - 11 = 0?

The solutions of x² - 12x - 11 = 0 are x = 6 + √47 or x = 6 - √47, that is x ≈ 12.855655 and x ≈ -0.8556546.

Does x² - 12x - 11 = 0 factor?

Not over the integers, because the discriminant 188 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 12x - 11?

(6, -47). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 12, product = c/a = -11.

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    Solve x² - 12x - 11 = 0: x = 6 + √47 or x = 6 - √47 (steps) | WordGPT