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Solve 3x² + 9x = 0

The solutions of 3x² + 9x = 0 are x = 0 or x = -3.

x² +x += 0
3x2+9x=0⇒x=0, −33x^2 + 9x = 0 \Rightarrow x = 0,\ -3
Discriminant D = 81 · x ≈ 0, -3

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

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Graph of y = 3x² + 9x-5-4-3-2-112-1010203040x = 0x = -3vertexy = 3x² + 9x
Graph of y = 3x² + 9x. The curve crosses the x-axis at x = 0 and x = -3; the vertex is at (-1.5, -6.75).

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 = 9, c = 0.

    3x2+9x=03x^2 + 9x = 0
  2. 2

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

    D=92−4⋅3⋅0=81+0=81D = 9^2 - 4\cdot3\cdot0 = 81 + 0 = 81
  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+3)=03\left(x\right)\left(x + 3\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+3=0⇒x=−3x = 0 \Rightarrow x = 0\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±816x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{81}}{6}
  7. 7

    Simplify to get the solutions:

    x=0orx=−3x = 0\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 3·(0) + 9·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 9x = 0\ \Rightarrow\ x = 0,\ -3 \]
\[ 3x^2 + 9x = 0 \]
\[ D = 9^2 - 4\cdot3\cdot0 = 81 + 0 = 81 \]
\[ 3\left(x\right)\left(x + 3\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{81}}{6} \]
\[ x = 0\quad\text{or}\quad x = -3 \]

The parabola y = 3x² + 9x

xy
-4.520.25
-3.55.25
-2.5-3.75
-1.5-6.75
-0.5-3.75
0.55.25
1.520.25

Questions

What are the solutions of 3x² + 9x = 0?

The solutions of 3x² + 9x = 0 are x = 0 or x = -3.

Does 3x² + 9x = 0 factor?

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

What is the vertex of y = 3x² + 9x?

(-3/2, -27/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 0.

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