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Solve 2x² - 12x + 12 = 0

The solutions of 2x² - 12x + 12 = 0 are x = 3 + √3 or x = 3 - √3, that is x ≈ 4.7320508 and x ≈ 1.2679492.

x² +x += 0
2x2−12x+12=0⇒x=3+3, 3−32x^2 - 12x + 12 = 0 \Rightarrow x = 3 + \sqrt{3},\ 3 - \sqrt{3}
Discriminant D = 48 · x ≈ 4.73205, 1.26795

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

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Graph of y = 2x² - 12x + 12-11234567-10-551015202530x = 4.7320508x = 1.2679492vertexy = 2x² - 12x + 12
Graph of y = 2x² - 12x + 12. The curve crosses the x-axis at x = 4.7320508 and x = 1.2679492; the vertex is at (3, -6).

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 = 12.

    2x2−12x+12=02x^2 - 12x + 12 = 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⋅12=144−96=48D = (-12)^2 - 4\cdot2\cdot12 = 144 - 96 = 48
  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±484x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{48}}{4}
  5. 5

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

    x=3+3orx=3−3x = 3 + \sqrt{3}\quad\text{or}\quad x = 3 - \sqrt{3}
  6. 6

    As decimals: x ≈ 4.7320508 and x ≈ 1.2679492.

  7. 7

    Check numerically: for x ≈ 4.7320508 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 12 = 0\ \Rightarrow\ x = 3 + \sqrt{3},\ 3 - \sqrt{3} \]
\[ 2x^2 - 12x + 12 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot12 = 144 - 96 = 48 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{48}}{4} \]
\[ x = 3 + \sqrt{3}\quad\text{or}\quad x = 3 - \sqrt{3} \]

The parabola y = 2x² - 12x + 12

xy
012
12
2-4
3-6
4-4
52
612

Questions

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

The solutions of 2x² - 12x + 12 = 0 are x = 3 + √3 or x = 3 - √3, that is x ≈ 4.7320508 and x ≈ 1.2679492.

Does 2x² - 12x + 12 = 0 factor?

Not over the integers, because the discriminant 48 is not a perfect square. Use the quadratic formula.

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

(3, -6). 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 = 6.

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