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Solve 5x² + 8x = 0

The solutions of 5x² + 8x = 0 are x = 0 or x = -8/5.

x² +x += 0
5x2+8x=0⇒x=0, −855x^2 + 8x = 0 \Rightarrow x = 0,\ -\frac{8}{5}
Discriminant D = 64 · x ≈ 0, -1.6

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

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Graph of y = 5x² + 8x-4-3-2-112-101020304050x = 0x = -1.6vertexy = 5x² + 8x
Graph of y = 5x² + 8x. The curve crosses the x-axis at x = 0 and x = -1.6; the vertex is at (-0.8, -3.2).

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 = 8, c = 0.

    5x2+8x=05x^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⋅5⋅0=64+0=64D = 8^2 - 4\cdot5\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)(5x+8)=0\left(x\right)\left(5x + 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=05x+8=0⇒x=−85x = 0 \Rightarrow x = 0\qquad 5x + 8 = 0 \Rightarrow x = -\frac{8}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±6410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{64}}{10}
  7. 7

    Simplify to get the solutions:

    x=0orx=−85x = 0\quad\text{or}\quad x = -\frac{8}{5}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 5·(0) + 8·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 8x = 0\ \Rightarrow\ x = 0,\ -\frac{8}{5} \]
\[ 5x^2 + 8x = 0 \]
\[ D = 8^2 - 4\cdot5\cdot0 = 64 + 0 = 64 \]
\[ \left(x\right)\left(5x + 8\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 5x + 8 = 0 \Rightarrow x = -\frac{8}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{64}}{10} \]
\[ x = 0\quad\text{or}\quad x = -\frac{8}{5} \]

The parabola y = 5x² + 8x

xy
-3.841.8
-2.816.8
-1.81.8
-0.8-3.2
0.21.8
1.216.8
2.241.8

Questions

What are the solutions of 5x² + 8x = 0?

The solutions of 5x² + 8x = 0 are x = 0 or x = -8/5.

Does 5x² + 8x = 0 factor?

Yes: (x)(5x+8) = 0.

What is the vertex of y = 5x² + 8x?

(-4/5, -16/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8/5, product = c/a = 0.

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