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Solve 5x² - 5x - 11 = 0

The solutions of 5x² - 5x - 11 = 0 are x = (5 + 7√5)/10 or x = (5 - 7√5)/10, that is x ≈ 2.0652476 and x ≈ -1.0652476.

x² +x += 0
5x2−5x−11=0⇒x=5+7510, 5−75105x^2 - 5x - 11 = 0 \Rightarrow x = \frac{5 + 7\sqrt{5}}{10},\ \frac{5 - 7\sqrt{5}}{10}
Discriminant D = 245 · x ≈ 2.06525, -1.06525

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

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Graph of y = 5x² - 5x - 11-3-2-11234-20204060x = 2.0652476x = -1.0652476vertexy = 5x² - 5x - 11
Graph of y = 5x² - 5x - 11. The curve crosses the x-axis at x = 2.0652476 and x = -1.0652476; the vertex is at (0.5, -12.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -5, c = -11.

    5x2−5x−11=05x^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⋅5⋅(−11)=25+220=245D = (-5)^2 - 4\cdot5\cdot(-11) = 25 + 220 = 245
  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±24510x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{245}}{10}
  5. 5

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

    x=5+7510orx=5−7510x = \frac{5 + 7\sqrt{5}}{10}\quad\text{or}\quad x = \frac{5 - 7\sqrt{5}}{10}
  6. 6

    As decimals: x ≈ 2.0652476 and x ≈ -1.0652476.

  7. 7

    Check numerically: for x ≈ 2.0652476 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 5x - 11 = 0\ \Rightarrow\ x = \frac{5 + 7\sqrt{5}}{10},\ \frac{5 - 7\sqrt{5}}{10} \]
\[ 5x^2 - 5x - 11 = 0 \]
\[ D = (-5)^2 - 4\cdot5\cdot(-11) = 25 + 220 = 245 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{245}}{10} \]
\[ x = \frac{5 + 7\sqrt{5}}{10}\quad\text{or}\quad x = \frac{5 - 7\sqrt{5}}{10} \]

The parabola y = 5x² - 5x - 11

xy
-2.532.75
-1.57.75
-0.5-7.25
0.5-12.25
1.5-7.25
2.57.75
3.532.75

Questions

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

The solutions of 5x² - 5x - 11 = 0 are x = (5 + 7√5)/10 or x = (5 - 7√5)/10, that is x ≈ 2.0652476 and x ≈ -1.0652476.

Does 5x² - 5x - 11 = 0 factor?

Not over the integers, because the discriminant 245 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 5x² - 5x - 11?

(1/2, -49/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -11/5.

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    Solve 5x² - 5x - 11 = 0: x = (5 + 7√5)/10 or x = (5 - 7√5)/10 (steps) | WordGPT