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Solve 2x² - 6x - 1 = 0

The solutions of 2x² - 6x - 1 = 0 are x = (3 + √11)/2 or x = (3 - √11)/2, that is x ≈ 3.1583124 and x ≈ -0.1583124.

x² +x += 0
2x2−6x−1=0⇒x=3+112, 3−1122x^2 - 6x - 1 = 0 \Rightarrow x = \frac{3 + \sqrt{11}}{2},\ \frac{3 - \sqrt{11}}{2}
Discriminant D = 44 · x ≈ 3.15831, -0.158312

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

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Graph of y = 2x² - 6x - 1-2-112345-10-551015202530x = 3.1583124x = -0.1583124vertexy = 2x² - 6x - 1
Graph of y = 2x² - 6x - 1. The curve crosses the x-axis at x = 3.1583124 and x = -0.1583124; the vertex is at (1.5, -5.5).

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 = -6, c = -1.

    2x2−6x−1=02x^2 - 6x - 1 = 0
  2. 2

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

    D=(−6)2−4⋅2⋅(−1)=36+8=44D = (-6)^2 - 4\cdot2\cdot(-1) = 36 + 8 = 44
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=6±444x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{4}
  5. 5

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

    x=3+112orx=3−112x = \frac{3 + \sqrt{11}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{11}}{2}
  6. 6

    As decimals: x ≈ 3.1583124 and x ≈ -0.1583124.

  7. 7

    Check numerically: for x ≈ 3.1583124 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 6x - 1 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{11}}{2},\ \frac{3 - \sqrt{11}}{2} \]
\[ 2x^2 - 6x - 1 = 0 \]
\[ D = (-6)^2 - 4\cdot2\cdot(-1) = 36 + 8 = 44 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{44}}{4} \]
\[ x = \frac{3 + \sqrt{11}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{11}}{2} \]

The parabola y = 2x² - 6x - 1

xy
-1.512.5
-0.52.5
0.5-3.5
1.5-5.5
2.5-3.5
3.52.5
4.512.5

Questions

What are the solutions of 2x² - 6x - 1 = 0?

The solutions of 2x² - 6x - 1 = 0 are x = (3 + √11)/2 or x = (3 - √11)/2, that is x ≈ 3.1583124 and x ≈ -0.1583124.

Does 2x² - 6x - 1 = 0 factor?

Not over the integers, because the discriminant 44 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 6x - 1?

(3/2, -11/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -1/2.

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    Solve 2x² - 6x - 1 = 0: x = (3 + √11)/2 or x = (3 - √11)/2 (steps) | WordGPT