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Solve 3x² - 10 = 0

The solutions of 3x² - 10 = 0 are x = (√30)/3 or x = (-√30)/3, that is x ≈ 1.8257419 and x ≈ -1.8257419.

x² +x += 0
3x2−10=0⇒x=303, −3033x^2 - 10 = 0 \Rightarrow x = \frac{\sqrt{30}}{3},\ \frac{-\sqrt{30}}{3}
Discriminant D = 120 · x ≈ 1.82574, -1.82574

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

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Graph of y = 3x² - 10-4-3-2-11234-101020304050x = 1.8257419x = -1.8257419vertexy = 3x² - 10
Graph of y = 3x² - 10. The curve crosses the x-axis at x = 1.8257419 and x = -1.8257419; 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 = 3, b = 0, c = -10.

    3x2−10=03x^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⋅3⋅(−10)=0+120=120D = 0^2 - 4\cdot3\cdot(-10) = 0 + 120 = 120
  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±1206x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{120}}{6}
  5. 5

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

    x=303orx=−303x = \frac{\sqrt{30}}{3}\quad\text{or}\quad x = \frac{-\sqrt{30}}{3}
  6. 6

    As decimals: x ≈ 1.8257419 and x ≈ -1.8257419.

  7. 7

    Check numerically: for x ≈ 1.8257419 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 10 = 0\ \Rightarrow\ x = \frac{\sqrt{30}}{3},\ \frac{-\sqrt{30}}{3} \]
\[ 3x^2 - 10 = 0 \]
\[ D = 0^2 - 4\cdot3\cdot(-10) = 0 + 120 = 120 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{120}}{6} \]
\[ x = \frac{\sqrt{30}}{3}\quad\text{or}\quad x = \frac{-\sqrt{30}}{3} \]

The parabola y = 3x² - 10

xy
-317
-22
-1-7
0-10
1-7
22
317

Questions

What are the solutions of 3x² - 10 = 0?

The solutions of 3x² - 10 = 0 are x = (√30)/3 or x = (-√30)/3, that is x ≈ 1.8257419 and x ≈ -1.8257419.

Does 3x² - 10 = 0 factor?

Not over the integers, because the discriminant 120 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 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 = -10/3.

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    Solve 3x² - 10 = 0: x = (√30)/3 or x = (-√30)/3 (steps) | WordGPT