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Solve -x² + 2x = 0

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

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

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

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Graph of y = -x² + 2x-2-11234-12-10-8-6-4-22x = 0x = 2vertexy = -x² + 2x
Graph of y = -x² + 2x. The curve crosses the x-axis at x = 0 and x = 2; the vertex is at (1, 1).

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 = 0.

    −x2+2x=0-x^2 + 2x = 0
  2. 2

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

    D=22−4⋅(−1)⋅0=4+0=4D = 2^2 - 4\cdot(-1)\cdot0 = 4 + 0 = 4
  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)(x−2)=0-\left(x\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=0⇒x=0x−2=0⇒x=2x = 0 \Rightarrow x = 0\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

    x=0orx=2x = 0\quad\text{or}\quad x = 2
  8. 8

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

LaTeX source of the formulas on this page
\[ -x^2 + 2x = 0\ \Rightarrow\ x = 0,\ 2 \]
\[ -x^2 + 2x = 0 \]
\[ D = 2^2 - 4\cdot(-1)\cdot0 = 4 + 0 = 4 \]
\[ -\left(x\right)\left(x - 2\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{4}}{-2} \]
\[ x = 0\quad\text{or}\quad x = 2 \]

The parabola y = -x² + 2x

xy
-2-8
-1-3
00
11
20
3-3
4-8

Questions

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

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

Does -x² + 2x = 0 factor?

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

What is the vertex of y = -x² + 2x?

(1, 1). 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 = 0.

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