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Solve -2x² - 2x + 2 = 0

The solutions of -2x² - 2x + 2 = 0 are x = (-1 + √5)/2 or x = (-1 - √5)/2, that is x ≈ -1.618034 and x ≈ 0.61803399.

x² +x += 0
−2x2−2x+2=0⇒x=−1+52, −1−52-2x^2 - 2x + 2 = 0 \Rightarrow x = \frac{-1 + \sqrt{5}}{2},\ \frac{-1 - \sqrt{5}}{2}
Discriminant D = 20 · x ≈ -1.61803, 0.618034

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

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Graph of y = -2x² - 2x + 2-4-3-2-1123-25-20-15-10-55x = -1.618034x = 0.61803399vertexy = -2x² - 2x + 2
Graph of y = -2x² - 2x + 2. The curve crosses the x-axis at x = -1.618034 and x = 0.61803399; the vertex is at (-0.5, 2.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = -2, c = 2.

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

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

    D=(−2)2−4⋅(−2)⋅2=4+16=20D = (-2)^2 - 4\cdot(-2)\cdot2 = 4 + 16 = 20
  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±20−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{20}}{-4}
  5. 5

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

    x=−1+52orx=−1−52x = \frac{-1 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{5}}{2}
  6. 6

    As decimals: x ≈ -1.618034 and x ≈ 0.61803399.

  7. 7

    Check numerically: for x ≈ -1.618034 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 2x + 2 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{5}}{2},\ \frac{-1 - \sqrt{5}}{2} \]
\[ -2x^2 - 2x + 2 = 0 \]
\[ D = (-2)^2 - 4\cdot(-2)\cdot2 = 4 + 16 = 20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{20}}{-4} \]
\[ x = \frac{-1 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{5}}{2} \]

The parabola y = -2x² - 2x + 2

xy
-3.5-15.5
-2.5-5.5
-1.50.5
-0.52.5
0.50.5
1.5-5.5
2.5-15.5

Questions

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

The solutions of -2x² - 2x + 2 = 0 are x = (-1 + √5)/2 or x = (-1 - √5)/2, that is x ≈ -1.618034 and x ≈ 0.61803399.

Does -2x² - 2x + 2 = 0 factor?

Not over the integers, because the discriminant 20 is not a perfect square. Use the quadratic formula.

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

(-1/2, 5/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = -1.

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    Solve -2x² - 2x + 2 = 0: x = (-1 + √5)/2 or x = (-1 - √5)/2 (steps) | WordGPT