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Solve 2x² - 7x + 3 = 0

The solutions of 2x² - 7x + 3 = 0 are x = 3 or x = 1/2.

x² +x += 0
2x2−7x+3=0⇒x=3, 122x^2 - 7x + 3 = 0 \Rightarrow x = 3,\ \frac{1}{2}
Discriminant D = 25 · x ≈ 3, 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 - 7x + 3-2-112345-5510152025x = 3x = 0.5vertexy = 2x² - 7x + 3
Graph of y = 2x² - 7x + 3. The curve crosses the x-axis at x = 3 and x = 0.5; the vertex is at (1.75, -3.125).

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 = -7, c = 3.

    2x2−7x+3=02x^2 - 7x + 3 = 0
  2. 2

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

    D=(−7)2−4⋅2⋅3=49−24=25D = (-7)^2 - 4\cdot2\cdot3 = 49 - 24 = 25
  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−3)(2x−1)=0\left(x - 3\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−3=0⇒x=32x−1=0⇒x=12x - 3 = 0 \Rightarrow x = 3\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±254x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{25}}{4}
  7. 7

    Simplify to get the solutions:

    x=3orx=12x = 3\quad\text{or}\quad x = \frac{1}{2}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 2·(9) + (-7)·(3) + 3 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 7x + 3 = 0\ \Rightarrow\ x = 3,\ \frac{1}{2} \]
\[ 2x^2 - 7x + 3 = 0 \]
\[ D = (-7)^2 - 4\cdot2\cdot3 = 49 - 24 = 25 \]
\[ \left(x - 3\right)\left(2x - 1\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{25}}{4} \]
\[ x = 3\quad\text{or}\quad x = \frac{1}{2} \]

The parabola y = 2x² - 7x + 3

xy
-1.2514.875
-0.254.875
0.75-1.125
1.75-3.125
2.75-1.125
3.754.875
4.7514.875

Questions

What are the solutions of 2x² - 7x + 3 = 0?

The solutions of 2x² - 7x + 3 = 0 are x = 3 or x = 1/2.

Does 2x² - 7x + 3 = 0 factor?

Yes: (x-3)(2x-1) = 0.

What is the vertex of y = 2x² - 7x + 3?

(7/4, -25/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7/2, product = c/a = 3/2.

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