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Solve 2x² - 12x + 3 = 0

The solutions of 2x² - 12x + 3 = 0 are x = (6 + √30)/2 or x = (6 - √30)/2, that is x ≈ 5.7386128 and x ≈ 0.26138721.

x² +x += 0
2x2−12x+3=0⇒x=6+302, 6−3022x^2 - 12x + 3 = 0 \Rightarrow x = \frac{6 + \sqrt{30}}{2},\ \frac{6 - \sqrt{30}}{2}
Discriminant D = 120 · x ≈ 5.73861, 0.261387

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

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Graph of y = 2x² - 12x + 3-2-112345678-20-1010203040x = 5.7386128x = 0.26138721vertexy = 2x² - 12x + 3
Graph of y = 2x² - 12x + 3. The curve crosses the x-axis at x = 5.7386128 and x = 0.26138721; the vertex is at (3, -15).

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 = -12, c = 3.

    2x2−12x+3=02x^2 - 12x + 3 = 0
  2. 2

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

    D=(−12)2−4⋅2⋅3=144−24=120D = (-12)^2 - 4\cdot2\cdot3 = 144 - 24 = 120
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±1204x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{120}}{4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=6+302orx=6−302x = \frac{6 + \sqrt{30}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{30}}{2}
  6. 6

    As decimals: x ≈ 5.7386128 and x ≈ 0.26138721.

  7. 7

    Check numerically: for x ≈ 5.7386128 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 3 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{30}}{2},\ \frac{6 - \sqrt{30}}{2} \]
\[ 2x^2 - 12x + 3 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot3 = 144 - 24 = 120 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{120}}{4} \]
\[ x = \frac{6 + \sqrt{30}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{30}}{2} \]

The parabola y = 2x² - 12x + 3

xy
03
1-7
2-13
3-15
4-13
5-7
63

Questions

What are the solutions of 2x² - 12x + 3 = 0?

The solutions of 2x² - 12x + 3 = 0 are x = (6 + √30)/2 or x = (6 - √30)/2, that is x ≈ 5.7386128 and x ≈ 0.26138721.

Does 2x² - 12x + 3 = 0 factor?

Not over the integers, because the discriminant 120 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 12x + 3?

(3, -15). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 3/2.

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    Solve 2x² - 12x + 3 = 0: x = (6 + √30)/2 or x = (6 - √30)/2 (steps) | WordGPT