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Solve -2x² - 6x + 1 = 0

The solutions of -2x² - 6x + 1 = 0 are x = (-3 + √11)/2 or x = (-3 - √11)/2, that is x ≈ -3.1583124 and x ≈ 0.1583124.

x² +x += 0
−2x2−6x+1=0⇒x=−3+112, −3−112-2x^2 - 6x + 1 = 0 \Rightarrow x = \frac{-3 + \sqrt{11}}{2},\ \frac{-3 - \sqrt{11}}{2}
Discriminant D = 44 · x ≈ -3.15831, 0.158312

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

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Graph of y = -2x² - 6x + 1-5-4-3-2-112-30-25-20-15-10-5510x = -3.1583124x = 0.1583124vertexy = -2x² - 6x + 1
Graph of y = -2x² - 6x + 1. The curve crosses the x-axis at x = -3.1583124 and x = 0.1583124; the vertex is at (-1.5, 5.5).

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 = -6, c = 1.

    −2x2−6x+1=0-2x^2 - 6x + 1 = 0
  2. 2

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

    D=(−6)2−4⋅(−2)⋅1=36+8=44D = (-6)^2 - 4\cdot(-2)\cdot1 = 36 + 8 = 44
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=6±44−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{-4}
  5. 5

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

    x=−3+112orx=−3−112x = \frac{-3 + \sqrt{11}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{11}}{2}
  6. 6

    As decimals: x ≈ -3.1583124 and x ≈ 0.1583124.

  7. 7

    Check numerically: for x ≈ -3.1583124 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 6x + 1 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{11}}{2},\ \frac{-3 - \sqrt{11}}{2} \]
\[ -2x^2 - 6x + 1 = 0 \]
\[ D = (-6)^2 - 4\cdot(-2)\cdot1 = 36 + 8 = 44 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{-4} \]
\[ x = \frac{-3 + \sqrt{11}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{11}}{2} \]

The parabola y = -2x² - 6x + 1

xy
-4.5-12.5
-3.5-2.5
-2.53.5
-1.55.5
-0.53.5
0.5-2.5
1.5-12.5

Questions

What are the solutions of -2x² - 6x + 1 = 0?

The solutions of -2x² - 6x + 1 = 0 are x = (-3 + √11)/2 or x = (-3 - √11)/2, that is x ≈ -3.1583124 and x ≈ 0.1583124.

Does -2x² - 6x + 1 = 0 factor?

Not over the integers, because the discriminant 44 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 6x + 1?

(-3/2, 11/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -1/2.

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    Solve -2x² - 6x + 1 = 0: x = (-3 + √11)/2 or x = (-3 - √11)/2 (steps) | WordGPT