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Solve 4x² - 8 = 0

The solutions of 4x² - 8 = 0 are x = √2 or x = -√2, that is x ≈ 1.4142136 and x ≈ -1.4142136.

x² +x += 0
4x2−8=0⇒x=2, −24x^2 - 8 = 0 \Rightarrow x = \sqrt{2},\ -\sqrt{2}
Discriminant D = 128 · x ≈ 1.41421, -1.41421

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

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Graph of y = 4x² - 8-3-2-1123-101020304050x = 1.4142136x = -1.4142136vertexy = 4x² - 8
Graph of y = 4x² - 8. The curve crosses the x-axis at x = 1.4142136 and x = -1.4142136; the vertex is at (0, -8).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = 0, c = -8.

    4x2−8=04x^2 - 8 = 0
  2. 2

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

    D=02−4⋅4⋅(−8)=0+128=128D = 0^2 - 4\cdot4\cdot(-8) = 0 + 128 = 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=0±1288x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{128}}{8}
  5. 5

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

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

    As decimals: x ≈ 1.4142136 and x ≈ -1.4142136.

  7. 7

    Check numerically: for x ≈ 1.4142136 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 8 = 0\ \Rightarrow\ x = \sqrt{2},\ -\sqrt{2} \]
\[ 4x^2 - 8 = 0 \]
\[ D = 0^2 - 4\cdot4\cdot(-8) = 0 + 128 = 128 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{128}}{8} \]
\[ x = \sqrt{2}\quad\text{or}\quad x = -\sqrt{2} \]

The parabola y = 4x² - 8

xy
-328
-28
-1-4
0-8
1-4
28
328

Questions

What are the solutions of 4x² - 8 = 0?

The solutions of 4x² - 8 = 0 are x = √2 or x = -√2, that is x ≈ 1.4142136 and x ≈ -1.4142136.

Does 4x² - 8 = 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 = 4x² - 8?

(0, -8). 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 = -2.

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