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Solve 2x² + 11x = 0

The solutions of 2x² + 11x = 0 are x = 0 or x = -11/2.

x² +x += 0
2x2+11x=0⇒x=0, −1122x^2 + 11x = 0 \Rightarrow x = 0,\ -\frac{11}{2}
Discriminant D = 121 · x ≈ 0, -5.5

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

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Graph of y = 2x² + 11x-8-6-4-22-20-1010203040x = 0x = -5.5vertexy = 2x² + 11x
Graph of y = 2x² + 11x. The curve crosses the x-axis at x = 0 and x = -5.5; the vertex is at (-2.75, -15.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = 11, c = 0.

    2x2+11x=02x^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⋅2⋅0=121+0=121D = 11^2 - 4\cdot2\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)(2x+11)=0\left(x\right)\left(2x + 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=02x+11=0⇒x=−112x = 0 \Rightarrow x = 0\qquad 2x + 11 = 0 \Rightarrow x = -\frac{11}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1214x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{121}}{4}
  7. 7

    Simplify to get the solutions:

    x=0orx=−112x = 0\quad\text{or}\quad x = -\frac{11}{2}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = 2·(0) + 11·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 11x = 0\ \Rightarrow\ x = 0,\ -\frac{11}{2} \]
\[ 2x^2 + 11x = 0 \]
\[ D = 11^2 - 4\cdot2\cdot0 = 121 + 0 = 121 \]
\[ \left(x\right)\left(2x + 11\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 2x + 11 = 0 \Rightarrow x = -\frac{11}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{121}}{4} \]
\[ x = 0\quad\text{or}\quad x = -\frac{11}{2} \]

The parabola y = 2x² + 11x

xy
-5.752.875
-4.75-7.125
-3.75-13.125
-2.75-15.125
-1.75-13.125
-0.75-7.125
0.252.875

Questions

What are the solutions of 2x² + 11x = 0?

The solutions of 2x² + 11x = 0 are x = 0 or x = -11/2.

Does 2x² + 11x = 0 factor?

Yes: (x)(2x+11) = 0.

What is the vertex of y = 2x² + 11x?

(-11/4, -121/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/2, product = c/a = 0.

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