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Solve 5x² - 10 = 0

The solutions of 5x² - 10 = 0 are x = √2 or x = -√2, that is x ≈ 1.4142136 and x ≈ -1.4142136.

x² +x += 0
5x2−10=0⇒x=2, −25x^2 - 10 = 0 \Rightarrow x = \sqrt{2},\ -\sqrt{2}
Discriminant D = 200 · 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 = 5x² - 10-3-2-1123-20204060x = 1.4142136x = -1.4142136vertexy = 5x² - 10
Graph of y = 5x² - 10. The curve crosses the x-axis at x = 1.4142136 and x = -1.4142136; the vertex is at (0, -10).

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 = -10.

    5x2−10=05x^2 - 10 = 0
  2. 2

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

    D=02−4⋅5⋅(−10)=0+200=200D = 0^2 - 4\cdot5\cdot(-10) = 0 + 200 = 200
  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±20010x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{200}}{10}
  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
\[ 5x^2 - 10 = 0\ \Rightarrow\ x = \sqrt{2},\ -\sqrt{2} \]
\[ 5x^2 - 10 = 0 \]
\[ D = 0^2 - 4\cdot5\cdot(-10) = 0 + 200 = 200 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{200}}{10} \]
\[ x = \sqrt{2}\quad\text{or}\quad x = -\sqrt{2} \]

The parabola y = 5x² - 10

xy
-335
-210
-1-5
0-10
1-5
210
335

Questions

What are the solutions of 5x² - 10 = 0?

The solutions of 5x² - 10 = 0 are x = √2 or x = -√2, that is x ≈ 1.4142136 and x ≈ -1.4142136.

Does 5x² - 10 = 0 factor?

Not over the integers, because the discriminant 200 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 5x² - 10?

(0, -10). 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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