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Solve -2x² - 3x + 6 = 0

The solutions of -2x² - 3x + 6 = 0 are x = (-3 + √57)/4 or x = (-3 - √57)/4, that is x ≈ -2.6374586 and x ≈ 1.1374586.

x² +x += 0
−2x2−3x+6=0⇒x=−3+574, −3−574-2x^2 - 3x + 6 = 0 \Rightarrow x = \frac{-3 + \sqrt{57}}{4},\ \frac{-3 - \sqrt{57}}{4}
Discriminant D = 57 · x ≈ -2.63746, 1.13746

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

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Graph of y = -2x² - 3x + 6-5-4-3-2-1123-30-20-1010x = -2.6374586x = 1.1374586vertexy = -2x² - 3x + 6
Graph of y = -2x² - 3x + 6. The curve crosses the x-axis at x = -2.6374586 and x = 1.1374586; the vertex is at (-0.75, 7.125).

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 = -3, c = 6.

    −2x2−3x+6=0-2x^2 - 3x + 6 = 0
  2. 2

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

    D=(−3)2−4⋅(−2)⋅6=9+48=57D = (-3)^2 - 4\cdot(-2)\cdot6 = 9 + 48 = 57
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±57−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{57}}{-4}
  5. 5

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

    x=−3+574orx=−3−574x = \frac{-3 + \sqrt{57}}{4}\quad\text{or}\quad x = \frac{-3 - \sqrt{57}}{4}
  6. 6

    As decimals: x ≈ -2.6374586 and x ≈ 1.1374586.

  7. 7

    Check numerically: for x ≈ -2.6374586 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 3x + 6 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{57}}{4},\ \frac{-3 - \sqrt{57}}{4} \]
\[ -2x^2 - 3x + 6 = 0 \]
\[ D = (-3)^2 - 4\cdot(-2)\cdot6 = 9 + 48 = 57 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{57}}{-4} \]
\[ x = \frac{-3 + \sqrt{57}}{4}\quad\text{or}\quad x = \frac{-3 - \sqrt{57}}{4} \]

The parabola y = -2x² - 3x + 6

xy
-3.75-10.875
-2.75-0.875
-1.755.125
-0.757.125
0.255.125
1.25-0.875
2.25-10.875

Questions

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

The solutions of -2x² - 3x + 6 = 0 are x = (-3 + √57)/4 or x = (-3 - √57)/4, that is x ≈ -2.6374586 and x ≈ 1.1374586.

Does -2x² - 3x + 6 = 0 factor?

Not over the integers, because the discriminant 57 is not a perfect square. Use the quadratic formula.

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

(-3/4, 57/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

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

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    Solve -2x² - 3x + 6 = 0: x = (-3 + √57)/4 or x = (-3 - √57)/4 (steps) | WordGPT