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Solve -x² + 2x + 8 = 0

The solutions of -x² + 2x + 8 = 0 are x = -2 or x = 4.

x² +x += 0
−x2+2x+8=0⇒x=−2, 4-x^2 + 2x + 8 = 0 \Rightarrow x = -2,\ 4
Discriminant D = 36 · x ≈ -2, 4

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

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Graph of y = -x² + 2x + 8-4-2246-20-15-10-5510x = -2x = 4vertexy = -x² + 2x + 8
Graph of y = -x² + 2x + 8. The curve crosses the x-axis at x = -2 and x = 4; the vertex is at (1, 9).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 2, c = 8.

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

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

    D=22−4⋅(−1)⋅8=4+32=36D = 2^2 - 4\cdot(-1)\cdot8 = 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:

    −(x+2)(x−4)=0-\left(x + 2\right)\left(x - 4\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+2=0⇒x=−2x−4=0⇒x=4x + 2 = 0 \Rightarrow x = -2\qquad x - 4 = 0 \Rightarrow x = 4
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

    x=−2orx=4x = -2\quad\text{or}\quad x = 4
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = -1·(4) + 2·(-2) + 8 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 2x + 8 = 0\ \Rightarrow\ x = -2,\ 4 \]
\[ -x^2 + 2x + 8 = 0 \]
\[ D = 2^2 - 4\cdot(-1)\cdot8 = 4 + 32 = 36 \]
\[ -\left(x + 2\right)\left(x - 4\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x - 4 = 0 \Rightarrow x = 4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{36}}{-2} \]
\[ x = -2\quad\text{or}\quad x = 4 \]

The parabola y = -x² + 2x + 8

xy
-20
-15
08
19
28
35
40

Questions

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

The solutions of -x² + 2x + 8 = 0 are x = -2 or x = 4.

Does -x² + 2x + 8 = 0 factor?

Yes: -1(x+2)(x-4) = 0.

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

(1, 9). 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 = -8.

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