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Solve 4x² - 4x + 1 = 0

The solution of 4x² - 4x + 1 = 0 is x = 1/2 (a repeated root).

x² +x += 0
4x2−4x+1=0⇒x=124x^2 - 4x + 1 = 0 \Rightarrow x = \frac{1}{2}
Discriminant D = 0 · x ≈ 0.5

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

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Graph of y = 4x² - 4x + 1-2-1.5-1-0.50.511.522.53510152025x = 0.5vertexy = 4x² - 4x + 1
Graph of y = 4x² - 4x + 1. The curve crosses the x-axis at x = 0.5; the vertex is at (0.5, 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 = 4, b = -4, c = 1.

    4x2−4x+1=04x^2 - 4x + 1 = 0
  2. 2

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

    D=(−4)2−4⋅4⋅1=16−16=0D = (-4)^2 - 4\cdot4\cdot1 = 16 - 16 = 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:

    (2x−1)(2x−1)=0\left(2x - 1\right)\left(2x - 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:

    2x−1=0⇒x=122x - 1 = 0 \Rightarrow x = \frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±08x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{0}}{8}
  7. 7

    Simplify to get the solution:

    x=12x = \frac{1}{2}
  8. 8

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

LaTeX source of the formulas on this page
\[ 4x^2 - 4x + 1 = 0\ \Rightarrow\ x = \frac{1}{2} \]
\[ 4x^2 - 4x + 1 = 0 \]
\[ D = (-4)^2 - 4\cdot4\cdot1 = 16 - 16 = 0 \]
\[ \left(2x - 1\right)\left(2x - 1\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{0}}{8} \]
\[ x = \frac{1}{2} \]

The parabola y = 4x² - 4x + 1

xy
-2.536
-1.516
-0.54
0.50
1.54
2.516
3.536

Questions

What are the solutions of 4x² - 4x + 1 = 0?

The solution of 4x² - 4x + 1 = 0 is x = 1/2 (a repeated root).

Does 4x² - 4x + 1 = 0 factor?

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

What is the vertex of y = 4x² - 4x + 1?

(1/2, 0). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = 1/4.

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