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Solve x² - 11x - 4 = 0

The solutions of x² - 11x - 4 = 0 are x = (11 + √137)/2 or x = (11 - √137)/2, that is x ≈ 11.35235 and x ≈ -0.35234996.

x² +x += 0
x2−11x−4=0⇒x=11+1372, 11−1372x^2 - 11x - 4 = 0 \Rightarrow x = \frac{11 + \sqrt{137}}{2},\ \frac{11 - \sqrt{137}}{2}
Discriminant D = 137 · x ≈ 11.3523, -0.35235

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

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Graph of y = x² - 11x - 4-224681012-40-30-20-1010203040x = 11.35235x = -0.35234996vertexy = x² - 11x - 4
Graph of y = x² - 11x - 4. The curve crosses the x-axis at x = 11.35235 and x = -0.35234996; the vertex is at (5.5, -34.25).

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 = -4.

    x2−11x−4=0x^2 - 11x - 4 = 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⋅(−4)=121+16=137D = (-11)^2 - 4\cdot1\cdot(-4) = 121 + 16 = 137
  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±1372x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{137}}{2}
  5. 5

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

    x=11+1372orx=11−1372x = \frac{11 + \sqrt{137}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{137}}{2}
  6. 6

    As decimals: x ≈ 11.35235 and x ≈ -0.35234996.

  7. 7

    Check numerically: for x ≈ 11.35235 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 11x - 4 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{137}}{2},\ \frac{11 - \sqrt{137}}{2} \]
\[ x^2 - 11x - 4 = 0 \]
\[ D = (-11)^2 - 4\cdot1\cdot(-4) = 121 + 16 = 137 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{137}}{2} \]
\[ x = \frac{11 + \sqrt{137}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{137}}{2} \]

The parabola y = x² - 11x - 4

xy
2.5-25.25
3.5-30.25
4.5-33.25
5.5-34.25
6.5-33.25
7.5-30.25
8.5-25.25

Questions

What are the solutions of x² - 11x - 4 = 0?

The solutions of x² - 11x - 4 = 0 are x = (11 + √137)/2 or x = (11 - √137)/2, that is x ≈ 11.35235 and x ≈ -0.35234996.

Does x² - 11x - 4 = 0 factor?

Not over the integers, because the discriminant 137 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 11x - 4?

(11/2, -137/4). 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 = -4.

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    Solve x² - 11x - 4 = 0: x = (11 + √137)/2 or x = (11 - √137)/2 (steps) | WordGPT