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Solve -2x² - 4x + 6 = 0

The solutions of -2x² - 4x + 6 = 0 are x = -3 or x = 1.

x² +x += 0
−2x2−4x+6=0⇒x=−3, 1-2x^2 - 4x + 6 = 0 \Rightarrow x = -3,\ 1
Discriminant D = 64 · x ≈ -3, 1

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

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Graph of y = -2x² - 4x + 6-5-4-3-2-1123-30-20-1010x = -3x = 1vertexy = -2x² - 4x + 6
Graph of y = -2x² - 4x + 6. The curve crosses the x-axis at x = -3 and x = 1; the vertex is at (-1, 8).

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 = -4, c = 6.

    −2x2−4x+6=0-2x^2 - 4x + 6 = 0
  2. 2

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

    D=(−4)2−4⋅(−2)⋅6=16+48=64D = (-4)^2 - 4\cdot(-2)\cdot6 = 16 + 48 = 64
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −2(x+3)(x−1)=0-2\left(x + 3\right)\left(x - 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+3=0⇒x=−3x−1=0⇒x=1x + 3 = 0 \Rightarrow x = -3\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±64−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{64}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−3orx=1x = -3\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -3 back in: a·x² + b·x + c = -2·(9) + (-4)·(-3) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 4x + 6 = 0\ \Rightarrow\ x = -3,\ 1 \]
\[ -2x^2 - 4x + 6 = 0 \]
\[ D = (-4)^2 - 4\cdot(-2)\cdot6 = 16 + 48 = 64 \]
\[ -2\left(x + 3\right)\left(x - 1\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{64}}{-4} \]
\[ x = -3\quad\text{or}\quad x = 1 \]

The parabola y = -2x² - 4x + 6

xy
-4-10
-30
-26
-18
06
10
2-10

Questions

What are the solutions of -2x² - 4x + 6 = 0?

The solutions of -2x² - 4x + 6 = 0 are x = -3 or x = 1.

Does -2x² - 4x + 6 = 0 factor?

Yes: -2(x+3)(x-1) = 0.

What is the vertex of y = -2x² - 4x + 6?

(-1, 8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = -3.

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