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Solve 2x² - 12x + 2 = 0

The solutions of 2x² - 12x + 2 = 0 are x = 3 + 2√2 or x = 3 - 2√2, that is x ≈ 5.8284271 and x ≈ 0.17157288.

x² +x += 0
2x2−12x+2=0⇒x=3+22, 3−222x^2 - 12x + 2 = 0 \Rightarrow x = 3 + 2\sqrt{2},\ 3 - 2\sqrt{2}
Discriminant D = 128 · x ≈ 5.82843, 0.171573

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

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Graph of y = 2x² - 12x + 2-22468-20-1010203040x = 5.8284271x = 0.17157288vertexy = 2x² - 12x + 2
Graph of y = 2x² - 12x + 2. The curve crosses the x-axis at x = 5.8284271 and x = 0.17157288; the vertex is at (3, -16).

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 = 2.

    2x2−12x+2=02x^2 - 12x + 2 = 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⋅2=144−16=128D = (-12)^2 - 4\cdot2\cdot2 = 144 - 16 = 128
  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±1284x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{128}}{4}
  5. 5

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

    x=3+22orx=3−22x = 3 + 2\sqrt{2}\quad\text{or}\quad x = 3 - 2\sqrt{2}
  6. 6

    As decimals: x ≈ 5.8284271 and x ≈ 0.17157288.

  7. 7

    Check numerically: for x ≈ 5.8284271 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 2 = 0\ \Rightarrow\ x = 3 + 2\sqrt{2},\ 3 - 2\sqrt{2} \]
\[ 2x^2 - 12x + 2 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot2 = 144 - 16 = 128 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{128}}{4} \]
\[ x = 3 + 2\sqrt{2}\quad\text{or}\quad x = 3 - 2\sqrt{2} \]

The parabola y = 2x² - 12x + 2

xy
02
1-8
2-14
3-16
4-14
5-8
62

Questions

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

The solutions of 2x² - 12x + 2 = 0 are x = 3 + 2√2 or x = 3 - 2√2, that is x ≈ 5.8284271 and x ≈ 0.17157288.

Does 2x² - 12x + 2 = 0 factor?

Not over the integers, because the discriminant 128 is not a perfect square. Use the quadratic formula.

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

(3, -16). 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 = 1.

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