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Solve x² + 8x - 11 = 0

The solutions of x² + 8x - 11 = 0 are x = -4 + 3√3 or x = -4 - 3√3, that is x ≈ 1.1961524 and x ≈ -9.1961524.

x² +x += 0
x2+8x−11=0⇒x=−4+33, −4−33x^2 + 8x - 11 = 0 \Rightarrow x = -4 + 3\sqrt{3},\ -4 - 3\sqrt{3}
Discriminant D = 108 · x ≈ 1.19615, -9.19615

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

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Graph of y = x² + 8x - 11-10-8-6-4-22-30-20-10102030x = 1.1961524x = -9.1961524vertexy = x² + 8x - 11
Graph of y = x² + 8x - 11. The curve crosses the x-axis at x = 1.1961524 and x = -9.1961524; the vertex is at (-4, -27).

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 = 8, c = -11.

    x2+8x−11=0x^2 + 8x - 11 = 0
  2. 2

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

    D=82−4⋅1⋅(−11)=64+44=108D = 8^2 - 4\cdot1\cdot(-11) = 64 + 44 = 108
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±1082x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{108}}{2}
  5. 5

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

    x=−4+33orx=−4−33x = -4 + 3\sqrt{3}\quad\text{or}\quad x = -4 - 3\sqrt{3}
  6. 6

    As decimals: x ≈ 1.1961524 and x ≈ -9.1961524.

  7. 7

    Check numerically: for x ≈ 1.1961524 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 8x - 11 = 0\ \Rightarrow\ x = -4 + 3\sqrt{3},\ -4 - 3\sqrt{3} \]
\[ x^2 + 8x - 11 = 0 \]
\[ D = 8^2 - 4\cdot1\cdot(-11) = 64 + 44 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{108}}{2} \]
\[ x = -4 + 3\sqrt{3}\quad\text{or}\quad x = -4 - 3\sqrt{3} \]

The parabola y = x² + 8x - 11

xy
-7-18
-6-23
-5-26
-4-27
-3-26
-2-23
-1-18

Questions

What are the solutions of x² + 8x - 11 = 0?

The solutions of x² + 8x - 11 = 0 are x = -4 + 3√3 or x = -4 - 3√3, that is x ≈ 1.1961524 and x ≈ -9.1961524.

Does x² + 8x - 11 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 8x - 11?

(-4, -27). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8, product = c/a = -11.

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    Solve x² + 8x - 11 = 0: x = -4 + 3√3 or x = -4 - 3√3 (steps) | WordGPT