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Solve 5x² - 8 = 0

The solutions of 5x² - 8 = 0 are x = (2√10)/5 or x = (-2√10)/5, that is x ≈ 1.2649111 and x ≈ -1.2649111.

x² +x += 0
5x2−8=0⇒x=2105, −21055x^2 - 8 = 0 \Rightarrow x = \frac{2\sqrt{10}}{5},\ \frac{-2\sqrt{10}}{5}
Discriminant D = 160 · x ≈ 1.26491, -1.26491

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

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Graph of y = 5x² - 8-3-2-1123-10102030405060x = 1.2649111x = -1.2649111vertexy = 5x² - 8
Graph of y = 5x² - 8. The curve crosses the x-axis at x = 1.2649111 and x = -1.2649111; 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 = 5, b = 0, c = -8.

    5x2−8=05x^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⋅5⋅(−8)=0+160=160D = 0^2 - 4\cdot5\cdot(-8) = 0 + 160 = 160
  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±16010x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{160}}{10}
  5. 5

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

    x=2105orx=−2105x = \frac{2\sqrt{10}}{5}\quad\text{or}\quad x = \frac{-2\sqrt{10}}{5}
  6. 6

    As decimals: x ≈ 1.2649111 and x ≈ -1.2649111.

  7. 7

    Check numerically: for x ≈ 1.2649111 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 8 = 0\ \Rightarrow\ x = \frac{2\sqrt{10}}{5},\ \frac{-2\sqrt{10}}{5} \]
\[ 5x^2 - 8 = 0 \]
\[ D = 0^2 - 4\cdot5\cdot(-8) = 0 + 160 = 160 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{160}}{10} \]
\[ x = \frac{2\sqrt{10}}{5}\quad\text{or}\quad x = \frac{-2\sqrt{10}}{5} \]

The parabola y = 5x² - 8

xy
-337
-212
-1-3
0-8
1-3
212
337

Questions

What are the solutions of 5x² - 8 = 0?

The solutions of 5x² - 8 = 0 are x = (2√10)/5 or x = (-2√10)/5, that is x ≈ 1.2649111 and x ≈ -1.2649111.

Does 5x² - 8 = 0 factor?

Not over the integers, because the discriminant 160 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 5x² - 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 = -8/5.

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