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Solve x² - 11x + 2 = 0

The solutions of x² - 11x + 2 = 0 are x = (11 + √113)/2 or x = (11 - √113)/2, that is x ≈ 10.815073 and x ≈ 0.18492709.

x² +x += 0
x2−11x+2=0⇒x=11+1132, 11−1132x^2 - 11x + 2 = 0 \Rightarrow x = \frac{11 + \sqrt{113}}{2},\ \frac{11 - \sqrt{113}}{2}
Discriminant D = 113 · x ≈ 10.8151, 0.184927

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

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Graph of y = x² - 11x + 2-224681012-30-20-10102030x = 10.815073x = 0.18492709vertexy = x² - 11x + 2
Graph of y = x² - 11x + 2. The curve crosses the x-axis at x = 10.815073 and x = 0.18492709; the vertex is at (5.5, -28.25).

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 = 2.

    x2−11x+2=0x^2 - 11x + 2 = 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⋅2=121−8=113D = (-11)^2 - 4\cdot1\cdot2 = 121 - 8 = 113
  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±1132x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{113}}{2}
  5. 5

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

    x=11+1132orx=11−1132x = \frac{11 + \sqrt{113}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{113}}{2}
  6. 6

    As decimals: x ≈ 10.815073 and x ≈ 0.18492709.

  7. 7

    Check numerically: for x ≈ 10.815073 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 11x + 2 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{113}}{2},\ \frac{11 - \sqrt{113}}{2} \]
\[ x^2 - 11x + 2 = 0 \]
\[ D = (-11)^2 - 4\cdot1\cdot2 = 121 - 8 = 113 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{113}}{2} \]
\[ x = \frac{11 + \sqrt{113}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{113}}{2} \]

The parabola y = x² - 11x + 2

xy
2.5-19.25
3.5-24.25
4.5-27.25
5.5-28.25
6.5-27.25
7.5-24.25
8.5-19.25

Questions

What are the solutions of x² - 11x + 2 = 0?

The solutions of x² - 11x + 2 = 0 are x = (11 + √113)/2 or x = (11 - √113)/2, that is x ≈ 10.815073 and x ≈ 0.18492709.

Does x² - 11x + 2 = 0 factor?

Not over the integers, because the discriminant 113 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 11x + 2?

(11/2, -113/4). 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 = 2.

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    Solve x² - 11x + 2 = 0: x = (11 + √113)/2 or x = (11 - √113)/2 (steps) | WordGPT