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Solve 3x² + 6x = 0

The solutions of 3x² + 6x = 0 are x = 0 or x = -2.

x² +x += 0
3x2+6x=0⇒x=0, −23x^2 + 6x = 0 \Rightarrow x = 0,\ -2
Discriminant D = 36 · x ≈ 0, -2

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

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Graph of y = 3x² + 6x-4-3-2-112-55101520253035x = 0x = -2vertexy = 3x² + 6x
Graph of y = 3x² + 6x. The curve crosses the x-axis at x = 0 and x = -2; the vertex is at (-1, -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 = 3, b = 6, c = 0.

    3x2+6x=03x^2 + 6x = 0
  2. 2

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

    D=62−4⋅3⋅0=36+0=36D = 6^2 - 4\cdot3\cdot0 = 36 + 0 = 36
  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)(x+2)=03\left(x\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=0⇒x=0x+2=0⇒x=−2x = 0 \Rightarrow x = 0\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−6±366x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{36}}{6}
  7. 7

    Simplify to get the solutions:

    x=0orx=−2x = 0\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 3·(0) + 6·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 6x = 0\ \Rightarrow\ x = 0,\ -2 \]
\[ 3x^2 + 6x = 0 \]
\[ D = 6^2 - 4\cdot3\cdot0 = 36 + 0 = 36 \]
\[ 3\left(x\right)\left(x + 2\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{36}}{6} \]
\[ x = 0\quad\text{or}\quad x = -2 \]

The parabola y = 3x² + 6x

xy
-424
-39
-20
-1-3
00
19
224

Questions

What are the solutions of 3x² + 6x = 0?

The solutions of 3x² + 6x = 0 are x = 0 or x = -2.

Does 3x² + 6x = 0 factor?

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

What is the vertex of y = 3x² + 6x?

(-1, -3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = 0.

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