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Solve 5x² + 11x = 0

The solutions of 5x² + 11x = 0 are x = 0 or x = -11/5.

x² +x += 0
5x2+11x=0⇒x=0, −1155x^2 + 11x = 0 \Rightarrow x = 0,\ -\frac{11}{5}
Discriminant D = 121 · x ≈ 0, -2.2

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

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Graph of y = 5x² + 11x-4-3-2-112-10102030405060x = 0x = -2.2vertexy = 5x² + 11x
Graph of y = 5x² + 11x. The curve crosses the x-axis at x = 0 and x = -2.2; the vertex is at (-1.1, -6.05).

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 = 11, c = 0.

    5x2+11x=05x^2 + 11x = 0
  2. 2

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

    D=112−4⋅5⋅0=121+0=121D = 11^2 - 4\cdot5\cdot0 = 121 + 0 = 121
  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+11)=0\left(x\right)\left(5x + 11\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+11=0⇒x=−115x = 0 \Rightarrow x = 0\qquad 5x + 11 = 0 \Rightarrow x = -\frac{11}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±12110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{121}}{10}
  7. 7

    Simplify to get the solutions:

    x=0orx=−115x = 0\quad\text{or}\quad x = -\frac{11}{5}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 5·(0) + 11·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 11x = 0\ \Rightarrow\ x = 0,\ -\frac{11}{5} \]
\[ 5x^2 + 11x = 0 \]
\[ D = 11^2 - 4\cdot5\cdot0 = 121 + 0 = 121 \]
\[ \left(x\right)\left(5x + 11\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 5x + 11 = 0 \Rightarrow x = -\frac{11}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{121}}{10} \]
\[ x = 0\quad\text{or}\quad x = -\frac{11}{5} \]

The parabola y = 5x² + 11x

xy
-4.138.95
-3.113.95
-2.1-1.05
-1.1-6.05
-0.1-1.05
0.913.95
1.938.95

Questions

What are the solutions of 5x² + 11x = 0?

The solutions of 5x² + 11x = 0 are x = 0 or x = -11/5.

Does 5x² + 11x = 0 factor?

Yes: (x)(5x+11) = 0.

What is the vertex of y = 5x² + 11x?

(-11/10, -121/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/5, product = c/a = 0.

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