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Solve -2x² + 10x - 11 = 0

The solutions of -2x² + 10x - 11 = 0 are x = (5 + √3)/2 or x = (5 - √3)/2, that is x ≈ 1.6339746 and x ≈ 3.3660254.

x² +x += 0
−2x2+10x−11=0⇒x=5+32, 5−32-2x^2 + 10x - 11 = 0 \Rightarrow x = \frac{5 + \sqrt{3}}{2},\ \frac{5 - \sqrt{3}}{2}
Discriminant D = 12 · x ≈ 1.63397, 3.36603

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

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Graph of y = -2x² + 10x - 1112345-20-15-10-5x = 1.6339746x = 3.3660254vertexy = -2x² + 10x - 11
Graph of y = -2x² + 10x - 11. The curve crosses the x-axis at x = 1.6339746 and x = 3.3660254; the vertex is at (2.5, 1.5).

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 = 10, c = -11.

    −2x2+10x−11=0-2x^2 + 10x - 11 = 0
  2. 2

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

    D=102−4⋅(−2)⋅(−11)=100−88=12D = 10^2 - 4\cdot(-2)\cdot(-11) = 100 - 88 = 12
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±12−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{12}}{-4}
  5. 5

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

    x=5+32orx=5−32x = \frac{5 + \sqrt{3}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{3}}{2}
  6. 6

    As decimals: x ≈ 1.6339746 and x ≈ 3.3660254.

  7. 7

    Check numerically: for x ≈ 1.6339746 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 10x - 11 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{3}}{2},\ \frac{5 - \sqrt{3}}{2} \]
\[ -2x^2 + 10x - 11 = 0 \]
\[ D = 10^2 - 4\cdot(-2)\cdot(-11) = 100 - 88 = 12 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{12}}{-4} \]
\[ x = \frac{5 + \sqrt{3}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{3}}{2} \]

The parabola y = -2x² + 10x - 11

xy
-0.5-16.5
0.5-6.5
1.5-0.5
2.51.5
3.5-0.5
4.5-6.5
5.5-16.5

Questions

What are the solutions of -2x² + 10x - 11 = 0?

The solutions of -2x² + 10x - 11 = 0 are x = (5 + √3)/2 or x = (5 - √3)/2, that is x ≈ 1.6339746 and x ≈ 3.3660254.

Does -2x² + 10x - 11 = 0 factor?

Not over the integers, because the discriminant 12 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 10x - 11?

(5/2, 3/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = 11/2.

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