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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, 2-3x^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 + 12-4-3-2-11234-50-40-30-20-101020x = -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=0-3x^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\cdot(-3)\cdot12 = 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)=0-3\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±144−6x = \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
\[ -3x^2 + 12 = 0\ \Rightarrow\ x = -2,\ 2 \]
\[ -3x^2 + 12 = 0 \]
\[ D = 0^2 - 4\cdot(-3)\cdot12 = 0 + 144 = 144 \]
\[ -3\left(x + 2\right)\left(x - 2\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{144}}{-6} \]
\[ x = -2\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 12

xy
-3-15
-20
-19
012
19
20
3-15

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 downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -4.

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