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Solve -2x² - 8x + 3 = 0

The solutions of -2x² - 8x + 3 = 0 are x = (-4 + √22)/2 or x = (-4 - √22)/2, that is x ≈ -4.3452079 and x ≈ 0.34520788.

x² +x += 0
−2x2−8x+3=0⇒x=−4+222, −4−222-2x^2 - 8x + 3 = 0 \Rightarrow x = \frac{-4 + \sqrt{22}}{2},\ \frac{-4 - \sqrt{22}}{2}
Discriminant D = 88 · x ≈ -4.34521, 0.345208

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

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Graph of y = -2x² - 8x + 3-6-5-4-3-2-112-30-20-1010x = -4.3452079x = 0.34520788vertexy = -2x² - 8x + 3
Graph of y = -2x² - 8x + 3. The curve crosses the x-axis at x = -4.3452079 and x = 0.34520788; the vertex is at (-2, 11).

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 = -8, c = 3.

    −2x2−8x+3=0-2x^2 - 8x + 3 = 0
  2. 2

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

    D=(−8)2−4⋅(−2)⋅3=64+24=88D = (-8)^2 - 4\cdot(-2)\cdot3 = 64 + 24 = 88
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=8±88−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{88}}{-4}
  5. 5

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

    x=−4+222orx=−4−222x = \frac{-4 + \sqrt{22}}{2}\quad\text{or}\quad x = \frac{-4 - \sqrt{22}}{2}
  6. 6

    As decimals: x ≈ -4.3452079 and x ≈ 0.34520788.

  7. 7

    Check numerically: for x ≈ -4.3452079 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 8x + 3 = 0\ \Rightarrow\ x = \frac{-4 + \sqrt{22}}{2},\ \frac{-4 - \sqrt{22}}{2} \]
\[ -2x^2 - 8x + 3 = 0 \]
\[ D = (-8)^2 - 4\cdot(-2)\cdot3 = 64 + 24 = 88 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{88}}{-4} \]
\[ x = \frac{-4 + \sqrt{22}}{2}\quad\text{or}\quad x = \frac{-4 - \sqrt{22}}{2} \]

The parabola y = -2x² - 8x + 3

xy
-5-7
-43
-39
-211
-19
03
1-7

Questions

What are the solutions of -2x² - 8x + 3 = 0?

The solutions of -2x² - 8x + 3 = 0 are x = (-4 + √22)/2 or x = (-4 - √22)/2, that is x ≈ -4.3452079 and x ≈ 0.34520788.

Does -2x² - 8x + 3 = 0 factor?

Not over the integers, because the discriminant 88 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 8x + 3?

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

What are the sum and product of the roots?

Sum = −b/a = -4, product = c/a = -3/2.

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    Solve -2x² - 8x + 3 = 0: x = (-4 + √22)/2 or x = (-4 - √22)/2 (steps) | WordGPT