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Solve 2x² - 12x + 4 = 0

The solutions of 2x² - 12x + 4 = 0 are x = 3 + √7 or x = 3 - √7, that is x ≈ 5.6457513 and x ≈ 0.35424869.

x² +x += 0
2x2−12x+4=0⇒x=3+7, 3−72x^2 - 12x + 4 = 0 \Rightarrow x = 3 + \sqrt{7},\ 3 - \sqrt{7}
Discriminant D = 112 · x ≈ 5.64575, 0.354249

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

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Graph of y = 2x² - 12x + 4-2-112345678-20-1010203040x = 5.6457513x = 0.35424869vertexy = 2x² - 12x + 4
Graph of y = 2x² - 12x + 4. The curve crosses the x-axis at x = 5.6457513 and x = 0.35424869; the vertex is at (3, -14).

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 = 4.

    2x2−12x+4=02x^2 - 12x + 4 = 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⋅4=144−32=112D = (-12)^2 - 4\cdot2\cdot4 = 144 - 32 = 112
  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±1124x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{112}}{4}
  5. 5

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

    x=3+7orx=3−7x = 3 + \sqrt{7}\quad\text{or}\quad x = 3 - \sqrt{7}
  6. 6

    As decimals: x ≈ 5.6457513 and x ≈ 0.35424869.

  7. 7

    Check numerically: for x ≈ 5.6457513 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 4 = 0\ \Rightarrow\ x = 3 + \sqrt{7},\ 3 - \sqrt{7} \]
\[ 2x^2 - 12x + 4 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot4 = 144 - 32 = 112 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{112}}{4} \]
\[ x = 3 + \sqrt{7}\quad\text{or}\quad x = 3 - \sqrt{7} \]

The parabola y = 2x² - 12x + 4

xy
04
1-6
2-12
3-14
4-12
5-6
64

Questions

What are the solutions of 2x² - 12x + 4 = 0?

The solutions of 2x² - 12x + 4 = 0 are x = 3 + √7 or x = 3 - √7, that is x ≈ 5.6457513 and x ≈ 0.35424869.

Does 2x² - 12x + 4 = 0 factor?

Not over the integers, because the discriminant 112 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 12x + 4?

(3, -14). 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 = 2.

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    Solve 2x² - 12x + 4 = 0: x = 3 + √7 or x = 3 - √7 (steps) | WordGPT