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Solve -2x² + 2x + 4 = 0

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

x² +x += 0
−2x2+2x+4=0⇒x=−1, 2-2x^2 + 2x + 4 = 0 \Rightarrow x = -1,\ 2
Discriminant D = 36 · x ≈ -1, 2

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

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Graph of y = -2x² + 2x + 4-3-2-11234-25-20-15-10-55x = -1x = 2vertexy = -2x² + 2x + 4
Graph of y = -2x² + 2x + 4. The curve crosses the x-axis at x = -1 and x = 2; the vertex is at (0.5, 4.5).

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 = 2, c = 4.

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

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

    D=22−4⋅(−2)⋅4=4+32=36D = 2^2 - 4\cdot(-2)\cdot4 = 4 + 32 = 36
  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+1)(x−2)=0-2\left(x + 1\right)\left(x - 2\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+1=0⇒x=−1x−2=0⇒x=2x + 1 = 0 \Rightarrow x = -1\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−2±36−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{36}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−1orx=2x = -1\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -2·(1) + 2·(-1) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 2x + 4 = 0\ \Rightarrow\ x = -1,\ 2 \]
\[ -2x^2 + 2x + 4 = 0 \]
\[ D = 2^2 - 4\cdot(-2)\cdot4 = 4 + 32 = 36 \]
\[ -2\left(x + 1\right)\left(x - 2\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{36}}{-4} \]
\[ x = -1\quad\text{or}\quad x = 2 \]

The parabola y = -2x² + 2x + 4

xy
-2.5-13.5
-1.5-3.5
-0.52.5
0.54.5
1.52.5
2.5-3.5
3.5-13.5

Questions

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

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

Does -2x² + 2x + 4 = 0 factor?

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

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

(1/2, 9/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -2.

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