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Solve x² - 11x - 8 = 0

The solutions of x² - 11x - 8 = 0 are x = (11 + 3√17)/2 or x = (11 - 3√17)/2, that is x ≈ 11.684658 and x ≈ -0.68465844.

x² +x += 0
x2−11x−8=0⇒x=11+3172, 11−3172x^2 - 11x - 8 = 0 \Rightarrow x = \frac{11 + 3\sqrt{17}}{2},\ \frac{11 - 3\sqrt{17}}{2}
Discriminant D = 153 · x ≈ 11.6847, -0.684658

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

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Graph of y = x² - 11x - 8-22468101214-40-202040x = 11.684658x = -0.68465844vertexy = x² - 11x - 8
Graph of y = x² - 11x - 8. The curve crosses the x-axis at x = 11.684658 and x = -0.68465844; the vertex is at (5.5, -38.25).

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 = -8.

    x2−11x−8=0x^2 - 11x - 8 = 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⋅(−8)=121+32=153D = (-11)^2 - 4\cdot1\cdot(-8) = 121 + 32 = 153
  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±1532x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{153}}{2}
  5. 5

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

    x=11+3172orx=11−3172x = \frac{11 + 3\sqrt{17}}{2}\quad\text{or}\quad x = \frac{11 - 3\sqrt{17}}{2}
  6. 6

    As decimals: x ≈ 11.684658 and x ≈ -0.68465844.

  7. 7

    Check numerically: for x ≈ 11.684658 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 11x - 8 = 0\ \Rightarrow\ x = \frac{11 + 3\sqrt{17}}{2},\ \frac{11 - 3\sqrt{17}}{2} \]
\[ x^2 - 11x - 8 = 0 \]
\[ D = (-11)^2 - 4\cdot1\cdot(-8) = 121 + 32 = 153 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{153}}{2} \]
\[ x = \frac{11 + 3\sqrt{17}}{2}\quad\text{or}\quad x = \frac{11 - 3\sqrt{17}}{2} \]

The parabola y = x² - 11x - 8

xy
2.5-29.25
3.5-34.25
4.5-37.25
5.5-38.25
6.5-37.25
7.5-34.25
8.5-29.25

Questions

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

The solutions of x² - 11x - 8 = 0 are x = (11 + 3√17)/2 or x = (11 - 3√17)/2, that is x ≈ 11.684658 and x ≈ -0.68465844.

Does x² - 11x - 8 = 0 factor?

Not over the integers, because the discriminant 153 is not a perfect square. Use the quadratic formula.

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

(11/2, -153/4). 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 = -8.

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    Solve x² - 11x - 8 = 0: x = (11 + 3√17)/2 or x = (11 - 3√17)/2 (steps) | WordGPT