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Solve x² - 5x - 11 = 0

The solutions of x² - 5x - 11 = 0 are x = (5 + √69)/2 or x = (5 - √69)/2, that is x ≈ 6.6533119 and x ≈ -1.6533119.

x² +x += 0
x2−5x−11=0⇒x=5+692, 5−692x^2 - 5x - 11 = 0 \Rightarrow x = \frac{5 + \sqrt{69}}{2},\ \frac{5 - \sqrt{69}}{2}
Discriminant D = 69 · x ≈ 6.65331, -1.65331

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

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Graph of y = x² - 5x - 11-4-22468-20-10102030x = 6.6533119x = -1.6533119vertexy = x² - 5x - 11
Graph of y = x² - 5x - 11. The curve crosses the x-axis at x = 6.6533119 and x = -1.6533119; the vertex is at (2.5, -17.25).

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 = -5, c = -11.

    x2−5x−11=0x^2 - 5x - 11 = 0
  2. 2

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

    D=(−5)2−4⋅1⋅(−11)=25+44=69D = (-5)^2 - 4\cdot1\cdot(-11) = 25 + 44 = 69
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=5±692x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{69}}{2}
  5. 5

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

    x=5+692orx=5−692x = \frac{5 + \sqrt{69}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{69}}{2}
  6. 6

    As decimals: x ≈ 6.6533119 and x ≈ -1.6533119.

  7. 7

    Check numerically: for x ≈ 6.6533119 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 5x - 11 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{69}}{2},\ \frac{5 - \sqrt{69}}{2} \]
\[ x^2 - 5x - 11 = 0 \]
\[ D = (-5)^2 - 4\cdot1\cdot(-11) = 25 + 44 = 69 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{69}}{2} \]
\[ x = \frac{5 + \sqrt{69}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{69}}{2} \]

The parabola y = x² - 5x - 11

xy
-0.5-8.25
0.5-13.25
1.5-16.25
2.5-17.25
3.5-16.25
4.5-13.25
5.5-8.25

Questions

What are the solutions of x² - 5x - 11 = 0?

The solutions of x² - 5x - 11 = 0 are x = (5 + √69)/2 or x = (5 - √69)/2, that is x ≈ 6.6533119 and x ≈ -1.6533119.

Does x² - 5x - 11 = 0 factor?

Not over the integers, because the discriminant 69 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 5x - 11?

(5/2, -69/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = -11.

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    Solve x² - 5x - 11 = 0: x = (5 + √69)/2 or x = (5 - √69)/2 (steps) | WordGPT