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Solve x² + 9x - 11 = 0

The solutions of x² + 9x - 11 = 0 are x = (-9 + 5√5)/2 or x = (-9 - 5√5)/2, that is x ≈ 1.0901699 and x ≈ -10.09017.

x² +x += 0
x2+9x−11=0⇒x=−9+552, −9−552x^2 + 9x - 11 = 0 \Rightarrow x = \frac{-9 + 5\sqrt{5}}{2},\ \frac{-9 - 5\sqrt{5}}{2}
Discriminant D = 125 · x ≈ 1.09017, -10.0902

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

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Graph of y = x² + 9x - 11-12-10-8-6-4-22-30-20-10102030x = 1.0901699x = -10.09017vertexy = x² + 9x - 11
Graph of y = x² + 9x - 11. The curve crosses the x-axis at x = 1.0901699 and x = -10.09017; the vertex is at (-4.5, -31.25).

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=0x^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=125D = 9^2 - 4\cdot1\cdot(-11) = 81 + 44 = 125
  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±1252x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{125}}{2}
  5. 5

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

    x=−9+552orx=−9−552x = \frac{-9 + 5\sqrt{5}}{2}\quad\text{or}\quad x = \frac{-9 - 5\sqrt{5}}{2}
  6. 6

    As decimals: x ≈ 1.0901699 and x ≈ -10.09017.

  7. 7

    Check numerically: for x ≈ 1.0901699 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 9x - 11 = 0\ \Rightarrow\ x = \frac{-9 + 5\sqrt{5}}{2},\ \frac{-9 - 5\sqrt{5}}{2} \]
\[ x^2 + 9x - 11 = 0 \]
\[ D = 9^2 - 4\cdot1\cdot(-11) = 81 + 44 = 125 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{125}}{2} \]
\[ x = \frac{-9 + 5\sqrt{5}}{2}\quad\text{or}\quad x = \frac{-9 - 5\sqrt{5}}{2} \]

The parabola y = x² + 9x - 11

xy
-7.5-22.25
-6.5-27.25
-5.5-30.25
-4.5-31.25
-3.5-30.25
-2.5-27.25
-1.5-22.25

Questions

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

The solutions of x² + 9x - 11 = 0 are x = (-9 + 5√5)/2 or x = (-9 - 5√5)/2, that is x ≈ 1.0901699 and x ≈ -10.09017.

Does x² + 9x - 11 = 0 factor?

Not over the integers, because the discriminant 125 is not a perfect square. Use the quadratic formula.

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

(-9/2, -125/4). The parabola opens upwards, so this is the minimum.

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 + 5√5)/2 or x = (-9 - 5√5)/2 (steps) | WordGPT