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Solve 2x² - 5x - 11 = 0

The solutions of 2x² - 5x - 11 = 0 are x = (5 + √113)/4 or x = (5 - √113)/4, that is x ≈ 3.9075365 and x ≈ -1.4075365.

x² +x += 0
2x2−5x−11=0⇒x=5+1134, 5−11342x^2 - 5x - 11 = 0 \Rightarrow x = \frac{5 + \sqrt{113}}{4},\ \frac{5 - \sqrt{113}}{4}
Discriminant D = 113 · x ≈ 3.90754, -1.40754

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

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Graph of y = 2x² - 5x - 11-3-2-1123456-20-1010203040x = 3.9075365x = -1.4075365vertexy = 2x² - 5x - 11
Graph of y = 2x² - 5x - 11. The curve crosses the x-axis at x = 3.9075365 and x = -1.4075365; the vertex is at (1.25, -14.125).

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 = -5, c = -11.

    2x2−5x−11=02x^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⋅2⋅(−11)=25+88=113D = (-5)^2 - 4\cdot2\cdot(-11) = 25 + 88 = 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=5±1134x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{113}}{4}
  5. 5

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

    x=5+1134orx=5−1134x = \frac{5 + \sqrt{113}}{4}\quad\text{or}\quad x = \frac{5 - \sqrt{113}}{4}
  6. 6

    As decimals: x ≈ 3.9075365 and x ≈ -1.4075365.

  7. 7

    Check numerically: for x ≈ 3.9075365 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 5x - 11 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{113}}{4},\ \frac{5 - \sqrt{113}}{4} \]
\[ 2x^2 - 5x - 11 = 0 \]
\[ D = (-5)^2 - 4\cdot2\cdot(-11) = 25 + 88 = 113 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{113}}{4} \]
\[ x = \frac{5 + \sqrt{113}}{4}\quad\text{or}\quad x = \frac{5 - \sqrt{113}}{4} \]

The parabola y = 2x² - 5x - 11

xy
-1.753.875
-0.75-6.125
0.25-12.125
1.25-14.125
2.25-12.125
3.25-6.125
4.253.875

Questions

What are the solutions of 2x² - 5x - 11 = 0?

The solutions of 2x² - 5x - 11 = 0 are x = (5 + √113)/4 or x = (5 - √113)/4, that is x ≈ 3.9075365 and x ≈ -1.4075365.

Does 2x² - 5x - 11 = 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 = 2x² - 5x - 11?

(5/4, -113/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/2, product = c/a = -11/2.

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    Solve 2x² - 5x - 11 = 0: x = (5 + √113)/4 or x = (5 - √113)/4 (steps) | WordGPT