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Solve 3x² - 8 = 0

The solutions of 3x² - 8 = 0 are x = (2√6)/3 or x = (-2√6)/3, that is x ≈ 1.6329932 and x ≈ -1.6329932.

x² +x += 0
3x2−8=0⇒x=263, −2633x^2 - 8 = 0 \Rightarrow x = \frac{2\sqrt{6}}{3},\ \frac{-2\sqrt{6}}{3}
Discriminant D = 96 · x ≈ 1.63299, -1.63299

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

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Graph of y = 3x² - 8-4-3-2-11234-1010203040x = 1.6329932x = -1.6329932vertexy = 3x² - 8
Graph of y = 3x² - 8. The curve crosses the x-axis at x = 1.6329932 and x = -1.6329932; 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 = 3, b = 0, c = -8.

    3x2−8=03x^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⋅3⋅(−8)=0+96=96D = 0^2 - 4\cdot3\cdot(-8) = 0 + 96 = 96
  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±966x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{96}}{6}
  5. 5

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

    x=263orx=−263x = \frac{2\sqrt{6}}{3}\quad\text{or}\quad x = \frac{-2\sqrt{6}}{3}
  6. 6

    As decimals: x ≈ 1.6329932 and x ≈ -1.6329932.

  7. 7

    Check numerically: for x ≈ 1.6329932 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 8 = 0\ \Rightarrow\ x = \frac{2\sqrt{6}}{3},\ \frac{-2\sqrt{6}}{3} \]
\[ 3x^2 - 8 = 0 \]
\[ D = 0^2 - 4\cdot3\cdot(-8) = 0 + 96 = 96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{96}}{6} \]
\[ x = \frac{2\sqrt{6}}{3}\quad\text{or}\quad x = \frac{-2\sqrt{6}}{3} \]

The parabola y = 3x² - 8

xy
-319
-24
-1-5
0-8
1-5
24
319

Questions

What are the solutions of 3x² - 8 = 0?

The solutions of 3x² - 8 = 0 are x = (2√6)/3 or x = (-2√6)/3, that is x ≈ 1.6329932 and x ≈ -1.6329932.

Does 3x² - 8 = 0 factor?

Not over the integers, because the discriminant 96 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 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 = -8/3.

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    Solve 3x² - 8 = 0: x = (2√6)/3 or x = (-2√6)/3 (steps) | WordGPT