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Solve 3x² + 10x = 0

The solutions of 3x² + 10x = 0 are x = 0 or x = -10/3.

x² +x += 0
3x2+10x=0⇒x=0, −1033x^2 + 10x = 0 \Rightarrow x = 0,\ -\frac{10}{3}
Discriminant D = 100 · x ≈ 0, -3.33333

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

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Graph of y = 3x² + 10x-5-4-3-2-112-1010203040x = 0x = -3.3333333vertexy = 3x² + 10x
Graph of y = 3x² + 10x. The curve crosses the x-axis at x = 0 and x = -3.3333333; the vertex is at (-1.6666667, -8.3333333).

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 = 10, c = 0.

    3x2+10x=03x^2 + 10x = 0
  2. 2

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

    D=102−4⋅3⋅0=100+0=100D = 10^2 - 4\cdot3\cdot0 = 100 + 0 = 100
  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+10)=0\left(x\right)\left(3x + 10\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+10=0⇒x=−103x = 0 \Rightarrow x = 0\qquad 3x + 10 = 0 \Rightarrow x = -\frac{10}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±1006x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{100}}{6}
  7. 7

    Simplify to get the solutions:

    x=0orx=−103x = 0\quad\text{or}\quad x = -\frac{10}{3}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 3·(0) + 10·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 10x = 0\ \Rightarrow\ x = 0,\ -\frac{10}{3} \]
\[ 3x^2 + 10x = 0 \]
\[ D = 10^2 - 4\cdot3\cdot0 = 100 + 0 = 100 \]
\[ \left(x\right)\left(3x + 10\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 3x + 10 = 0 \Rightarrow x = -\frac{10}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{100}}{6} \]
\[ x = 0\quad\text{or}\quad x = -\frac{10}{3} \]

The parabola y = 3x² + 10x

xy
-4.666718.667267
-3.66673.6670667
-2.6667-5.3331333
-1.6667-8.3333333
-0.6667-5.3335333
0.33333.6662667
1.333318.666067

Questions

What are the solutions of 3x² + 10x = 0?

The solutions of 3x² + 10x = 0 are x = 0 or x = -10/3.

Does 3x² + 10x = 0 factor?

Yes: (x)(3x+10) = 0.

What is the vertex of y = 3x² + 10x?

(-5/3, -25/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -10/3, product = c/a = 0.

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