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Solve x² + 11x + 11 = 0

The solutions of x² + 11x + 11 = 0 are x = (-11 + √77)/2 or x = (-11 - √77)/2, that is x ≈ -1.1125178 and x ≈ -9.8874822.

x² +x += 0
x2+11x+11=0⇒x=−11+772, −11−772x^2 + 11x + 11 = 0 \Rightarrow x = \frac{-11 + \sqrt{77}}{2},\ \frac{-11 - \sqrt{77}}{2}
Discriminant D = 77 · x ≈ -1.11252, -9.88748

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

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Graph of y = x² + 11x + 11-12-10-8-6-4-2-20-10102030x = -1.1125178x = -9.8874822vertexy = x² + 11x + 11
Graph of y = x² + 11x + 11. The curve crosses the x-axis at x = -1.1125178 and x = -9.8874822; the vertex is at (-5.5, -19.25).

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 = 11.

    x2+11x+11=0x^2 + 11x + 11 = 0
  2. 2

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

    D=112−4⋅1⋅11=121−44=77D = 11^2 - 4\cdot1\cdot11 = 121 - 44 = 77
  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±772x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{77}}{2}
  5. 5

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

    x=−11+772orx=−11−772x = \frac{-11 + \sqrt{77}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{77}}{2}
  6. 6

    As decimals: x ≈ -1.1125178 and x ≈ -9.8874822.

  7. 7

    Check numerically: for x ≈ -1.1125178 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 11x + 11 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{77}}{2},\ \frac{-11 - \sqrt{77}}{2} \]
\[ x^2 + 11x + 11 = 0 \]
\[ D = 11^2 - 4\cdot1\cdot11 = 121 - 44 = 77 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{77}}{2} \]
\[ x = \frac{-11 + \sqrt{77}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{77}}{2} \]

The parabola y = x² + 11x + 11

xy
-8.5-10.25
-7.5-15.25
-6.5-18.25
-5.5-19.25
-4.5-18.25
-3.5-15.25
-2.5-10.25

Questions

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

The solutions of x² + 11x + 11 = 0 are x = (-11 + √77)/2 or x = (-11 - √77)/2, that is x ≈ -1.1125178 and x ≈ -9.8874822.

Does x² + 11x + 11 = 0 factor?

Not over the integers, because the discriminant 77 is not a perfect square. Use the quadratic formula.

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

(-11/2, -77/4). 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 = 11.

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    Solve x² + 11x + 11 = 0: x = (-11 + √77)/2 or x = (-11 - √77)/2 (steps) | WordGPT