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Solve -x² + 8x + 11 = 0

The solutions of -x² + 8x + 11 = 0 are x = 4 + 3√3 or x = 4 - 3√3, that is x ≈ -1.1961524 and x ≈ 9.1961524.

x² +x += 0
−x2+8x+11=0⇒x=4+33, 4−33-x^2 + 8x + 11 = 0 \Rightarrow x = 4 + 3\sqrt{3},\ 4 - 3\sqrt{3}
Discriminant D = 108 · x ≈ -1.19615, 9.19615

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

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Graph of y = -x² + 8x + 11-2246810-30-20-10102030x = -1.1961524x = 9.1961524vertexy = -x² + 8x + 11
Graph of y = -x² + 8x + 11. The curve crosses the x-axis at x = -1.1961524 and x = 9.1961524; the vertex is at (4, 27).

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 = 8, c = 11.

    −x2+8x+11=0-x^2 + 8x + 11 = 0
  2. 2

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

    D=82−4⋅(−1)⋅11=64+44=108D = 8^2 - 4\cdot(-1)\cdot11 = 64 + 44 = 108
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±108−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{108}}{-2}
  5. 5

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

    x=4+33orx=4−33x = 4 + 3\sqrt{3}\quad\text{or}\quad x = 4 - 3\sqrt{3}
  6. 6

    As decimals: x ≈ -1.1961524 and x ≈ 9.1961524.

  7. 7

    Check numerically: for x ≈ -1.1961524 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x + 11 = 0\ \Rightarrow\ x = 4 + 3\sqrt{3},\ 4 - 3\sqrt{3} \]
\[ -x^2 + 8x + 11 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot11 = 64 + 44 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{108}}{-2} \]
\[ x = 4 + 3\sqrt{3}\quad\text{or}\quad x = 4 - 3\sqrt{3} \]

The parabola y = -x² + 8x + 11

xy
118
223
326
427
526
623
718

Questions

What are the solutions of -x² + 8x + 11 = 0?

The solutions of -x² + 8x + 11 = 0 are x = 4 + 3√3 or x = 4 - 3√3, that is x ≈ -1.1961524 and x ≈ 9.1961524.

Does -x² + 8x + 11 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 8x + 11?

(4, 27). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = -11.

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    Solve -x² + 8x + 11 = 0: x = 4 + 3√3 or x = 4 - 3√3 (steps) | WordGPT