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Solve 3x² + 8x = 0

The solutions of 3x² + 8x = 0 are x = 0 or x = -8/3.

x² +x += 0
3x2+8x=0⇒x=0, −833x^2 + 8x = 0 \Rightarrow x = 0,\ -\frac{8}{3}
Discriminant D = 64 · x ≈ 0, -2.66667

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

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Graph of y = 3x² + 8x-5-4-3-2-112-1010203040x = 0x = -2.6666667vertexy = 3x² + 8x
Graph of y = 3x² + 8x. The curve crosses the x-axis at x = 0 and x = -2.6666667; the vertex is at (-1.3333333, -5.3333333).

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 = 8, c = 0.

    3x2+8x=03x^2 + 8x = 0
  2. 2

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

    D=82−4⋅3⋅0=64+0=64D = 8^2 - 4\cdot3\cdot0 = 64 + 0 = 64
  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:

    (x)(3x+8)=0\left(x\right)\left(3x + 8\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=03x+8=0⇒x=−83x = 0 \Rightarrow x = 0\qquad 3x + 8 = 0 \Rightarrow x = -\frac{8}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±646x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{64}}{6}
  7. 7

    Simplify to get the solutions:

    x=0orx=−83x = 0\quad\text{or}\quad x = -\frac{8}{3}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 3·(0) + 8·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 8x = 0\ \Rightarrow\ x = 0,\ -\frac{8}{3} \]
\[ 3x^2 + 8x = 0 \]
\[ D = 8^2 - 4\cdot3\cdot0 = 64 + 0 = 64 \]
\[ \left(x\right)\left(3x + 8\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 3x + 8 = 0 \Rightarrow x = -\frac{8}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{64}}{6} \]
\[ x = 0\quad\text{or}\quad x = -\frac{8}{3} \]

The parabola y = 3x² + 8x

xy
-4.333321.666067
-3.33336.6662667
-2.3333-2.3335333
-1.3333-5.3333333
-0.3333-2.3331333
0.66676.6670667
1.666721.667267

Questions

What are the solutions of 3x² + 8x = 0?

The solutions of 3x² + 8x = 0 are x = 0 or x = -8/3.

Does 3x² + 8x = 0 factor?

Yes: (x)(3x+8) = 0.

What is the vertex of y = 3x² + 8x?

(-4/3, -16/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8/3, product = c/a = 0.

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