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Solve x² - 3 = 0

The solutions of x² - 3 = 0 are x = √3 or x = -√3, that is x ≈ 1.7320508 and x ≈ -1.7320508.

x² +x += 0
x2−3=0⇒x=3, −3x^2 - 3 = 0 \Rightarrow x = \sqrt{3},\ -\sqrt{3}
Discriminant D = 12 · x ≈ 1.73205, -1.73205

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

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Graph of y = x² - 3-4-3-2-11234-551015x = 1.7320508x = -1.7320508vertexy = x² - 3
Graph of y = x² - 3. The curve crosses the x-axis at x = 1.7320508 and x = -1.7320508; the vertex is at (0, -3).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 0, c = -3.

    x2−3=0x^2 - 3 = 0
  2. 2

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

    D=02−4⋅1⋅(−3)=0+12=12D = 0^2 - 4\cdot1\cdot(-3) = 0 + 12 = 12
  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±122x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{12}}{2}
  5. 5

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

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

    As decimals: x ≈ 1.7320508 and x ≈ -1.7320508.

  7. 7

    Check numerically: for x ≈ 1.7320508 the left side is -4.4408921e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 3 = 0\ \Rightarrow\ x = \sqrt{3},\ -\sqrt{3} \]
\[ x^2 - 3 = 0 \]
\[ D = 0^2 - 4\cdot1\cdot(-3) = 0 + 12 = 12 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{12}}{2} \]
\[ x = \sqrt{3}\quad\text{or}\quad x = -\sqrt{3} \]

The parabola y = x² - 3

xy
-36
-21
-1-2
0-3
1-2
21
36

Questions

What are the solutions of x² - 3 = 0?

The solutions of x² - 3 = 0 are x = √3 or x = -√3, that is x ≈ 1.7320508 and x ≈ -1.7320508.

Does x² - 3 = 0 factor?

Not over the integers, because the discriminant 12 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 3?

(0, -3). 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 = -3.

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