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Solve x² + 2x - 1 = 0

The solutions of x² + 2x - 1 = 0 are x = -1 + √2 or x = -1 - √2, that is x ≈ 0.41421356 and x ≈ -2.4142136.

x² +x += 0
x2+2x−1=0⇒x=−1+2, −1−2x^2 + 2x - 1 = 0 \Rightarrow x = -1 + \sqrt{2},\ -1 - \sqrt{2}
Discriminant D = 8 · x ≈ 0.414214, -2.41421

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

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Graph of y = x² + 2x - 1-4-3-2-112-4-22468101214x = 0.41421356x = -2.4142136vertexy = x² + 2x - 1
Graph of y = x² + 2x - 1. The curve crosses the x-axis at x = 0.41421356 and x = -2.4142136; the vertex is at (-1, -2).

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 = -1.

    x2+2x−1=0x^2 + 2x - 1 = 0
  2. 2

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

    D=22−4⋅1⋅(−1)=4+4=8D = 2^2 - 4\cdot1\cdot(-1) = 4 + 4 = 8
  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±82x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{8}}{2}
  5. 5

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

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

    As decimals: x ≈ 0.41421356 and x ≈ -2.4142136.

  7. 7

    Check numerically: for x ≈ 0.41421356 the left side is 2.220446e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 2x - 1 = 0\ \Rightarrow\ x = -1 + \sqrt{2},\ -1 - \sqrt{2} \]
\[ x^2 + 2x - 1 = 0 \]
\[ D = 2^2 - 4\cdot1\cdot(-1) = 4 + 4 = 8 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{8}}{2} \]
\[ x = -1 + \sqrt{2}\quad\text{or}\quad x = -1 - \sqrt{2} \]

The parabola y = x² + 2x - 1

xy
-47
-32
-2-1
-1-2
0-1
12
27

Questions

What are the solutions of x² + 2x - 1 = 0?

The solutions of x² + 2x - 1 = 0 are x = -1 + √2 or x = -1 - √2, that is x ≈ 0.41421356 and x ≈ -2.4142136.

Does x² + 2x - 1 = 0 factor?

Not over the integers, because the discriminant 8 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 2x - 1?

(-1, -2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = -1.

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