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Solve 3x² - 11 = 0

The solutions of 3x² - 11 = 0 are x = (√33)/3 or x = (-√33)/3, that is x ≈ 1.9148542 and x ≈ -1.9148542.

x² +x += 0
3x2−11=0⇒x=333, −3333x^2 - 11 = 0 \Rightarrow x = \frac{\sqrt{33}}{3},\ \frac{-\sqrt{33}}{3}
Discriminant D = 132 · x ≈ 1.91485, -1.91485

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

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Graph of y = 3x² - 11-4-3-2-11234-101020304050x = 1.9148542x = -1.9148542vertexy = 3x² - 11
Graph of y = 3x² - 11. The curve crosses the x-axis at x = 1.9148542 and x = -1.9148542; the vertex is at (0, -11).

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 = -11.

    3x2−11=03x^2 - 11 = 0
  2. 2

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

    D=02−4⋅3⋅(−11)=0+132=132D = 0^2 - 4\cdot3\cdot(-11) = 0 + 132 = 132
  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±1326x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{132}}{6}
  5. 5

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

    x=333orx=−333x = \frac{\sqrt{33}}{3}\quad\text{or}\quad x = \frac{-\sqrt{33}}{3}
  6. 6

    As decimals: x ≈ 1.9148542 and x ≈ -1.9148542.

  7. 7

    Check numerically: for x ≈ 1.9148542 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 11 = 0\ \Rightarrow\ x = \frac{\sqrt{33}}{3},\ \frac{-\sqrt{33}}{3} \]
\[ 3x^2 - 11 = 0 \]
\[ D = 0^2 - 4\cdot3\cdot(-11) = 0 + 132 = 132 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{132}}{6} \]
\[ x = \frac{\sqrt{33}}{3}\quad\text{or}\quad x = \frac{-\sqrt{33}}{3} \]

The parabola y = 3x² - 11

xy
-316
-21
-1-8
0-11
1-8
21
316

Questions

What are the solutions of 3x² - 11 = 0?

The solutions of 3x² - 11 = 0 are x = (√33)/3 or x = (-√33)/3, that is x ≈ 1.9148542 and x ≈ -1.9148542.

Does 3x² - 11 = 0 factor?

Not over the integers, because the discriminant 132 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 11?

(0, -11). 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 = -11/3.

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    Solve 3x² - 11 = 0: x = (√33)/3 or x = (-√33)/3 (steps) | WordGPT