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Solve -2x² - 6x - 3 = 0

The solutions of -2x² - 6x - 3 = 0 are x = (-3 + √3)/2 or x = (-3 - √3)/2, that is x ≈ -2.3660254 and x ≈ -0.6339746.

x² +x += 0
−2x2−6x−3=0⇒x=−3+32, −3−32-2x^2 - 6x - 3 = 0 \Rightarrow x = \frac{-3 + \sqrt{3}}{2},\ \frac{-3 - \sqrt{3}}{2}
Discriminant D = 12 · x ≈ -2.36603, -0.633975

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

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Graph of y = -2x² - 6x - 3-4-3-2-11-20-15-10-5x = -2.3660254x = -0.6339746vertexy = -2x² - 6x - 3
Graph of y = -2x² - 6x - 3. The curve crosses the x-axis at x = -2.3660254 and x = -0.6339746; the vertex is at (-1.5, 1.5).

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 = -3.

    −2x2−6x−3=0-2x^2 - 6x - 3 = 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)⋅(−3)=36−24=12D = (-6)^2 - 4\cdot(-2)\cdot(-3) = 36 - 24 = 12
  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±12−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{12}}{-4}
  5. 5

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

    x=−3+32orx=−3−32x = \frac{-3 + \sqrt{3}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{3}}{2}
  6. 6

    As decimals: x ≈ -2.3660254 and x ≈ -0.6339746.

  7. 7

    Check numerically: for x ≈ -2.3660254 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 6x - 3 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{3}}{2},\ \frac{-3 - \sqrt{3}}{2} \]
\[ -2x^2 - 6x - 3 = 0 \]
\[ D = (-6)^2 - 4\cdot(-2)\cdot(-3) = 36 - 24 = 12 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{12}}{-4} \]
\[ x = \frac{-3 + \sqrt{3}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{3}}{2} \]

The parabola y = -2x² - 6x - 3

xy
-4.5-16.5
-3.5-6.5
-2.5-0.5
-1.51.5
-0.5-0.5
0.5-6.5
1.5-16.5

Questions

What are the solutions of -2x² - 6x - 3 = 0?

The solutions of -2x² - 6x - 3 = 0 are x = (-3 + √3)/2 or x = (-3 - √3)/2, that is x ≈ -2.3660254 and x ≈ -0.6339746.

Does -2x² - 6x - 3 = 0 factor?

Not over the integers, because the discriminant 12 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 6x - 3?

(-3/2, 3/2). 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 = 3/2.

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