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Solve 2x² - 1 = 0

The solutions of 2x² - 1 = 0 are x = (√2)/2 or x = (-√2)/2, that is x ≈ 0.70710678 and x ≈ -0.70710678.

x² +x += 0
2x2−1=0⇒x=22, −222x^2 - 1 = 0 \Rightarrow x = \frac{\sqrt{2}}{2},\ \frac{-\sqrt{2}}{2}
Discriminant D = 8 · x ≈ 0.707107, -0.707107

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

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Graph of y = 2x² - 1-3-2-11235101520x = 0.70710678x = -0.70710678vertexy = 2x² - 1
Graph of y = 2x² - 1. The curve crosses the x-axis at x = 0.70710678 and x = -0.70710678; the vertex is at (0, -1).

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 = 0, c = -1.

    2x2−1=02x^2 - 1 = 0
  2. 2

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

    D=02−4⋅2⋅(−1)=0+8=8D = 0^2 - 4\cdot2\cdot(-1) = 0 + 8 = 8
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=0±84x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{8}}{4}
  5. 5

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

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

    As decimals: x ≈ 0.70710678 and x ≈ -0.70710678.

  7. 7

    Check numerically: for x ≈ 0.70710678 the left side is 2.220446e-16 (zero up to rounding) ✓

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

The parabola y = 2x² - 1

xy
-317
-27
-11
0-1
11
27
317

Questions

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

The solutions of 2x² - 1 = 0 are x = (√2)/2 or x = (-√2)/2, that is x ≈ 0.70710678 and x ≈ -0.70710678.

Does 2x² - 1 = 0 factor?

Not over the integers, because the discriminant 8 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 1?

(0, -1). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -1/2.

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