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Solve 3x² - 12 = 0

The solutions of 3x² - 12 = 0 are x = 2 or x = -2.

x² +x += 0
3x2−12=0⇒x=2, −23x^2 - 12 = 0 \Rightarrow x = 2,\ -2
Discriminant D = 144 · x ≈ 2, -2

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

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Graph of y = 3x² - 12-4-3-2-11234-20-101020304050x = 2x = -2vertexy = 3x² - 12
Graph of y = 3x² - 12. The curve crosses the x-axis at x = 2 and x = -2; the vertex is at (0, -12).

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 = -12.

    3x2−12=03x^2 - 12 = 0
  2. 2

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

    D=02−4⋅3⋅(−12)=0+144=144D = 0^2 - 4\cdot3\cdot(-12) = 0 + 144 = 144
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    3(x−2)(x+2)=03\left(x - 2\right)\left(x + 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−2=0⇒x=2x+2=0⇒x=−2x - 2 = 0 \Rightarrow x = 2\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=0±1446x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{144}}{6}
  7. 7

    Simplify to get the solutions:

    x=2orx=−2x = 2\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 3·(4) + 0·(2) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 12 = 0\ \Rightarrow\ x = 2,\ -2 \]
\[ 3x^2 - 12 = 0 \]
\[ D = 0^2 - 4\cdot3\cdot(-12) = 0 + 144 = 144 \]
\[ 3\left(x - 2\right)\left(x + 2\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{144}}{6} \]
\[ x = 2\quad\text{or}\quad x = -2 \]

The parabola y = 3x² - 12

xy
-315
-20
-1-9
0-12
1-9
20
315

Questions

What are the solutions of 3x² - 12 = 0?

The solutions of 3x² - 12 = 0 are x = 2 or x = -2.

Does 3x² - 12 = 0 factor?

Yes: 3(x-2)(x+2) = 0.

What is the vertex of y = 3x² - 12?

(0, -12). 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 = -4.

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