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Solve -x² + 2x + 11 = 0

The solutions of -x² + 2x + 11 = 0 are x = 1 + 2√3 or x = 1 - 2√3, that is x ≈ -2.4641016 and x ≈ 4.4641016.

x² +x += 0
−x2+2x+11=0⇒x=1+23, 1−23-x^2 + 2x + 11 = 0 \Rightarrow x = 1 + 2\sqrt{3},\ 1 - 2\sqrt{3}
Discriminant D = 48 · x ≈ -2.4641, 4.4641

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

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Graph of y = -x² + 2x + 11-4-2246-25-20-15-10-551015x = -2.4641016x = 4.4641016vertexy = -x² + 2x + 11
Graph of y = -x² + 2x + 11. The curve crosses the x-axis at x = -2.4641016 and x = 4.4641016; the vertex is at (1, 12).

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 = 2, c = 11.

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

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

    D=22−4⋅(−1)⋅11=4+44=48D = 2^2 - 4\cdot(-1)\cdot11 = 4 + 44 = 48
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−2±48−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{48}}{-2}
  5. 5

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

    x=1+23orx=1−23x = 1 + 2\sqrt{3}\quad\text{or}\quad x = 1 - 2\sqrt{3}
  6. 6

    As decimals: x ≈ -2.4641016 and x ≈ 4.4641016.

  7. 7

    Check numerically: for x ≈ -2.4641016 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 2x + 11 = 0\ \Rightarrow\ x = 1 + 2\sqrt{3},\ 1 - 2\sqrt{3} \]
\[ -x^2 + 2x + 11 = 0 \]
\[ D = 2^2 - 4\cdot(-1)\cdot11 = 4 + 44 = 48 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{48}}{-2} \]
\[ x = 1 + 2\sqrt{3}\quad\text{or}\quad x = 1 - 2\sqrt{3} \]

The parabola y = -x² + 2x + 11

xy
-23
-18
011
112
211
38
43

Questions

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

The solutions of -x² + 2x + 11 = 0 are x = 1 + 2√3 or x = 1 - 2√3, that is x ≈ -2.4641016 and x ≈ 4.4641016.

Does -x² + 2x + 11 = 0 factor?

Not over the integers, because the discriminant 48 is not a perfect square. Use the quadratic formula.

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

(1, 12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = -11.

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    Solve -x² + 2x + 11 = 0: x = 1 + 2√3 or x = 1 - 2√3 (steps) | WordGPT