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Solve 2x² - 12x + 11 = 0

The solutions of 2x² - 12x + 11 = 0 are x = (6 + √14)/2 or x = (6 - √14)/2, that is x ≈ 4.8708287 and x ≈ 1.1291713.

x² +x += 0
2x2−12x+11=0⇒x=6+142, 6−1422x^2 - 12x + 11 = 0 \Rightarrow x = \frac{6 + \sqrt{14}}{2},\ \frac{6 - \sqrt{14}}{2}
Discriminant D = 56 · x ≈ 4.87083, 1.12917

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

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Graph of y = 2x² - 12x + 11-11234567-10102030x = 4.8708287x = 1.1291713vertexy = 2x² - 12x + 11
Graph of y = 2x² - 12x + 11. The curve crosses the x-axis at x = 4.8708287 and x = 1.1291713; the vertex is at (3, -7).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -12, c = 11.

    2x2−12x+11=02x^2 - 12x + 11 = 0
  2. 2

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

    D=(−12)2−4⋅2⋅11=144−88=56D = (-12)^2 - 4\cdot2\cdot11 = 144 - 88 = 56
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±564x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{56}}{4}
  5. 5

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

    x=6+142orx=6−142x = \frac{6 + \sqrt{14}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{14}}{2}
  6. 6

    As decimals: x ≈ 4.8708287 and x ≈ 1.1291713.

  7. 7

    Check numerically: for x ≈ 4.8708287 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 11 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{14}}{2},\ \frac{6 - \sqrt{14}}{2} \]
\[ 2x^2 - 12x + 11 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot11 = 144 - 88 = 56 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{56}}{4} \]
\[ x = \frac{6 + \sqrt{14}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{14}}{2} \]

The parabola y = 2x² - 12x + 11

xy
011
11
2-5
3-7
4-5
51
611

Questions

What are the solutions of 2x² - 12x + 11 = 0?

The solutions of 2x² - 12x + 11 = 0 are x = (6 + √14)/2 or x = (6 - √14)/2, that is x ≈ 4.8708287 and x ≈ 1.1291713.

Does 2x² - 12x + 11 = 0 factor?

Not over the integers, because the discriminant 56 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 12x + 11?

(3, -7). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 11/2.

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    Solve 2x² - 12x + 11 = 0: x = (6 + √14)/2 or x = (6 - √14)/2 (steps) | WordGPT