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Solve 2x² - 12x + 1 = 0

The solutions of 2x² - 12x + 1 = 0 are x = (6 + √34)/2 or x = (6 - √34)/2, that is x ≈ 5.9154759 and x ≈ 0.084524053.

x² +x += 0
2x2−12x+1=0⇒x=6+342, 6−3422x^2 - 12x + 1 = 0 \Rightarrow x = \frac{6 + \sqrt{34}}{2},\ \frac{6 - \sqrt{34}}{2}
Discriminant D = 136 · x ≈ 5.91548, 0.0845241

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

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Graph of y = 2x² - 12x + 1-22468-20-1010203040x = 5.9154759x = 0.084524053vertexy = 2x² - 12x + 1
Graph of y = 2x² - 12x + 1. The curve crosses the x-axis at x = 5.9154759 and x = 0.084524053; the vertex is at (3, -17).

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 = 1.

    2x2−12x+1=02x^2 - 12x + 1 = 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⋅1=144−8=136D = (-12)^2 - 4\cdot2\cdot1 = 144 - 8 = 136
  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±1364x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{136}}{4}
  5. 5

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

    x=6+342orx=6−342x = \frac{6 + \sqrt{34}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{34}}{2}
  6. 6

    As decimals: x ≈ 5.9154759 and x ≈ 0.084524053.

  7. 7

    Check numerically: for x ≈ 5.9154759 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 1 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{34}}{2},\ \frac{6 - \sqrt{34}}{2} \]
\[ 2x^2 - 12x + 1 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot1 = 144 - 8 = 136 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{136}}{4} \]
\[ x = \frac{6 + \sqrt{34}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{34}}{2} \]

The parabola y = 2x² - 12x + 1

xy
01
1-9
2-15
3-17
4-15
5-9
61

Questions

What are the solutions of 2x² - 12x + 1 = 0?

The solutions of 2x² - 12x + 1 = 0 are x = (6 + √34)/2 or x = (6 - √34)/2, that is x ≈ 5.9154759 and x ≈ 0.084524053.

Does 2x² - 12x + 1 = 0 factor?

Not over the integers, because the discriminant 136 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 12x + 1?

(3, -17). 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 = 1/2.

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    Solve 2x² - 12x + 1 = 0: x = (6 + √34)/2 or x = (6 - √34)/2 (steps) | WordGPT