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Solve x² - 2x + 1 = 0

The solution of x² - 2x + 1 = 0 is x = 1 (a repeated root).

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

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

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Graph of y = x² - 2x + 1-1.5-1-0.50.511.522.533.5123456x = 1vertexy = x² - 2x + 1
Graph of y = x² - 2x + 1. The curve crosses the x-axis at x = 1; the vertex is at (1, 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 = 1, b = -2, c = 1.

    x2−2x+1=0x^2 - 2x + 1 = 0
  2. 2

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

    D=(−2)2−4⋅1⋅1=4−4=0D = (-2)^2 - 4\cdot1\cdot1 = 4 - 4 = 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:

    (x−1)(x−1)=0\left(x - 1\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−1=0⇒x=1x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=2±02x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{0}}{2}
  7. 7

    Simplify to get the solution:

    x=1x = 1
  8. 8

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

LaTeX source of the formulas on this page
\[ x^2 - 2x + 1 = 0\ \Rightarrow\ x = 1 \]
\[ x^2 - 2x + 1 = 0 \]
\[ D = (-2)^2 - 4\cdot1\cdot1 = 4 - 4 = 0 \]
\[ \left(x - 1\right)\left(x - 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{0}}{2} \]
\[ x = 1 \]

The parabola y = x² - 2x + 1

xy
-29
-14
01
10
21
34
49

Questions

What are the solutions of x² - 2x + 1 = 0?

The solution of x² - 2x + 1 = 0 is x = 1 (a repeated root).

Does x² - 2x + 1 = 0 factor?

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

What is the vertex of y = x² - 2x + 1?

(1, 0). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = 1.

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