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Solve -x² + 9x - 11 = 0

The solutions of -x² + 9x - 11 = 0 are x = (9 + √37)/2 or x = (9 - √37)/2, that is x ≈ 1.4586187 and x ≈ 7.5413813.

x² +x += 0
−x2+9x−11=0⇒x=9+372, 9−372-x^2 + 9x - 11 = 0 \Rightarrow x = \frac{9 + \sqrt{37}}{2},\ \frac{9 - \sqrt{37}}{2}
Discriminant D = 37 · x ≈ 1.45862, 7.54138

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

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Graph of y = -x² + 9x - 11246810-20-15-10-5510x = 1.4586187x = 7.5413813vertexy = -x² + 9x - 11
Graph of y = -x² + 9x - 11. The curve crosses the x-axis at x = 1.4586187 and x = 7.5413813; the vertex is at (4.5, 9.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 9, c = -11.

    −x2+9x−11=0-x^2 + 9x - 11 = 0
  2. 2

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

    D=92−4⋅(−1)⋅(−11)=81−44=37D = 9^2 - 4\cdot(-1)\cdot(-11) = 81 - 44 = 37
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±37−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{37}}{-2}
  5. 5

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

    x=9+372orx=9−372x = \frac{9 + \sqrt{37}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{37}}{2}
  6. 6

    As decimals: x ≈ 1.4586187 and x ≈ 7.5413813.

  7. 7

    Check numerically: for x ≈ 1.4586187 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 9x - 11 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{37}}{2},\ \frac{9 - \sqrt{37}}{2} \]
\[ -x^2 + 9x - 11 = 0 \]
\[ D = 9^2 - 4\cdot(-1)\cdot(-11) = 81 - 44 = 37 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{37}}{-2} \]
\[ x = \frac{9 + \sqrt{37}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{37}}{2} \]

The parabola y = -x² + 9x - 11

xy
1.50.25
2.55.25
3.58.25
4.59.25
5.58.25
6.55.25
7.50.25

Questions

What are the solutions of -x² + 9x - 11 = 0?

The solutions of -x² + 9x - 11 = 0 are x = (9 + √37)/2 or x = (9 - √37)/2, that is x ≈ 1.4586187 and x ≈ 7.5413813.

Does -x² + 9x - 11 = 0 factor?

Not over the integers, because the discriminant 37 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 9x - 11?

(9/2, 37/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 9, product = c/a = 11.

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    Solve -x² + 9x - 11 = 0: x = (9 + √37)/2 or x = (9 - √37)/2 (steps) | WordGPT