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Solve -2x² + 8x - 8 = 0

The solution of -2x² + 8x - 8 = 0 is x = 2 (a repeated root).

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

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

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Graph of y = -2x² + 8x - 8-0.50.511.522.533.544.5-12-10-8-6-4-2x = 2vertexy = -2x² + 8x - 8
Graph of y = -2x² + 8x - 8. The curve crosses the x-axis at x = 2; the vertex is at (2, 0).

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 = -8.

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

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

    D=82−4⋅(−2)⋅(−8)=64−64=0D = 8^2 - 4\cdot(-2)\cdot(-8) = 64 - 64 = 0
  3. 3

    D is zero, so there is exactly one solution (a repeated root).

  4. 4

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

    −2(x−2)(x−2)=0-2\left(x - 2\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−2=0⇒x=2x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±0−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{0}}{-4}
  7. 7

    Simplify to get the solution:

    x=2x = 2
  8. 8

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

LaTeX source of the formulas on this page
\[ -2x^2 + 8x - 8 = 0\ \Rightarrow\ x = 2 \]
\[ -2x^2 + 8x - 8 = 0 \]
\[ D = 8^2 - 4\cdot(-2)\cdot(-8) = 64 - 64 = 0 \]
\[ -2\left(x - 2\right)\left(x - 2\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{0}}{-4} \]
\[ x = 2 \]

The parabola y = -2x² + 8x - 8

xy
-1-18
0-8
1-2
20
3-2
4-8
5-18

Questions

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

The solution of -2x² + 8x - 8 = 0 is x = 2 (a repeated root).

Does -2x² + 8x - 8 = 0 factor?

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

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

(2, 0). 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 = 4.

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