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Solve 5x² + 3x = 0

The solutions of 5x² + 3x = 0 are x = 0 or x = -3/5.

x² +x += 0
5x2+3x=0⇒x=0, −355x^2 + 3x = 0 \Rightarrow x = 0,\ -\frac{3}{5}
Discriminant D = 9 · x ≈ 0, -0.6

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

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Graph of y = 5x² + 3x-3-2-11210203040x = 0x = -0.6vertexy = 5x² + 3x
Graph of y = 5x² + 3x. The curve crosses the x-axis at x = 0 and x = -0.6; the vertex is at (-0.3, -0.45).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = 3, c = 0.

    5x2+3x=05x^2 + 3x = 0
  2. 2

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

    D=32−4⋅5⋅0=9+0=9D = 3^2 - 4\cdot5\cdot0 = 9 + 0 = 9
  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)(5x+3)=0\left(x\right)\left(5x + 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=05x+3=0⇒x=−35x = 0 \Rightarrow x = 0\qquad 5x + 3 = 0 \Rightarrow x = -\frac{3}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−3±910x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{9}}{10}
  7. 7

    Simplify to get the solutions:

    x=0orx=−35x = 0\quad\text{or}\quad x = -\frac{3}{5}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 5·(0) + 3·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 3x = 0\ \Rightarrow\ x = 0,\ -\frac{3}{5} \]
\[ 5x^2 + 3x = 0 \]
\[ D = 3^2 - 4\cdot5\cdot0 = 9 + 0 = 9 \]
\[ \left(x\right)\left(5x + 3\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 5x + 3 = 0 \Rightarrow x = -\frac{3}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{9}}{10} \]
\[ x = 0\quad\text{or}\quad x = -\frac{3}{5} \]

The parabola y = 5x² + 3x

xy
-3.344.55
-2.319.55
-1.34.55
-0.3-0.45
0.74.55
1.719.55
2.744.55

Questions

What are the solutions of 5x² + 3x = 0?

The solutions of 5x² + 3x = 0 are x = 0 or x = -3/5.

Does 5x² + 3x = 0 factor?

Yes: (x)(5x+3) = 0.

What is the vertex of y = 5x² + 3x?

(-3/10, -9/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3/5, product = c/a = 0.

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