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Solve 2x² - 12x - 11 = 0

The solutions of 2x² - 12x - 11 = 0 are x = (6 + √58)/2 or x = (6 - √58)/2, that is x ≈ 6.8078866 and x ≈ -0.80788655.

x² +x += 0
2x2−12x−11=0⇒x=6+582, 6−5822x^2 - 12x - 11 = 0 \Rightarrow x = \frac{6 + \sqrt{58}}{2},\ \frac{6 - \sqrt{58}}{2}
Discriminant D = 232 · x ≈ 6.80789, -0.807887

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

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Graph of y = 2x² - 12x - 11-22468-202040x = 6.8078866x = -0.80788655vertexy = 2x² - 12x - 11
Graph of y = 2x² - 12x - 11. The curve crosses the x-axis at x = 6.8078866 and x = -0.80788655; the vertex is at (3, -29).

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=232D = (-12)^2 - 4\cdot2\cdot(-11) = 144 + 88 = 232
  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±2324x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{232}}{4}
  5. 5

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

    x=6+582orx=6−582x = \frac{6 + \sqrt{58}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{58}}{2}
  6. 6

    As decimals: x ≈ 6.8078866 and x ≈ -0.80788655.

  7. 7

    Check numerically: for x ≈ 6.8078866 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x - 11 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{58}}{2},\ \frac{6 - \sqrt{58}}{2} \]
\[ 2x^2 - 12x - 11 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot(-11) = 144 + 88 = 232 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{232}}{4} \]
\[ x = \frac{6 + \sqrt{58}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{58}}{2} \]

The parabola y = 2x² - 12x - 11

xy
0-11
1-21
2-27
3-29
4-27
5-21
6-11

Questions

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

The solutions of 2x² - 12x - 11 = 0 are x = (6 + √58)/2 or x = (6 - √58)/2, that is x ≈ 6.8078866 and x ≈ -0.80788655.

Does 2x² - 12x - 11 = 0 factor?

Not over the integers, because the discriminant 232 is not a perfect square. Use the quadratic formula.

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

(3, -29). 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 + √58)/2 or x = (6 - √58)/2 (steps) | WordGPT