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Solve 2x² - 3x + 1 = 0

The solutions of 2x² - 3x + 1 = 0 are x = 1 or x = 1/2.

x² +x += 0
2x2−3x+1=0⇒x=1, 122x^2 - 3x + 1 = 0 \Rightarrow x = 1,\ \frac{1}{2}
Discriminant D = 1 · x ≈ 1, 0.5

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

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Graph of y = 2x² - 3x + 1-2-1123-2246810121416x = 1x = 0.5vertexy = 2x² - 3x + 1
Graph of y = 2x² - 3x + 1. The curve crosses the x-axis at x = 1 and x = 0.5; the vertex is at (0.75, -0.125).

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 = -3, c = 1.

    2x2−3x+1=02x^2 - 3x + 1 = 0
  2. 2

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

    D=(−3)2−4⋅2⋅1=9−8=1D = (-3)^2 - 4\cdot2\cdot1 = 9 - 8 = 1
  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−1)(2x−1)=0\left(x - 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:

    x−1=0⇒x=12x−1=0⇒x=12x - 1 = 0 \Rightarrow x = 1\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±14x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{1}}{4}
  7. 7

    Simplify to get the solutions:

    x=1orx=12x = 1\quad\text{or}\quad x = \frac{1}{2}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 2·(1) + (-3)·(1) + 1 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 3x + 1 = 0\ \Rightarrow\ x = 1,\ \frac{1}{2} \]
\[ 2x^2 - 3x + 1 = 0 \]
\[ D = (-3)^2 - 4\cdot2\cdot1 = 9 - 8 = 1 \]
\[ \left(x - 1\right)\left(2x - 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{1}}{4} \]
\[ x = 1\quad\text{or}\quad x = \frac{1}{2} \]

The parabola y = 2x² - 3x + 1

xy
-2.2517.875
-1.257.875
-0.251.875
0.75-0.125
1.751.875
2.757.875
3.7517.875

Questions

What are the solutions of 2x² - 3x + 1 = 0?

The solutions of 2x² - 3x + 1 = 0 are x = 1 or x = 1/2.

Does 2x² - 3x + 1 = 0 factor?

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

What is the vertex of y = 2x² - 3x + 1?

(3/4, -1/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = 1/2.

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