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Solve 2x² + 11x + 9 = 0

The solutions of 2x² + 11x + 9 = 0 are x = -1 or x = -9/2.

x² +x += 0
2x2+11x+9=0⇒x=−1, −922x^2 + 11x + 9 = 0 \Rightarrow x = -1,\ -\frac{9}{2}
Discriminant D = 49 · x ≈ -1, -4.5

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

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Graph of y = 2x² + 11x + 9-7-6-5-4-3-2-11-10-551015202530x = -1x = -4.5vertexy = 2x² + 11x + 9
Graph of y = 2x² + 11x + 9. The curve crosses the x-axis at x = -1 and x = -4.5; the vertex is at (-2.75, -6.125).

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 = 9.

    2x2+11x+9=02x^2 + 11x + 9 = 0
  2. 2

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

    D=112−4⋅2⋅9=121−72=49D = 11^2 - 4\cdot2\cdot9 = 121 - 72 = 49
  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+1)(2x+9)=0\left(x + 1\right)\left(2x + 9\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+1=0⇒x=−12x+9=0⇒x=−92x + 1 = 0 \Rightarrow x = -1\qquad 2x + 9 = 0 \Rightarrow x = -\frac{9}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±494x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{4}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−92x = -1\quad\text{or}\quad x = -\frac{9}{2}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 2·(1) + 11·(-1) + 9 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 11x + 9 = 0\ \Rightarrow\ x = -1,\ -\frac{9}{2} \]
\[ 2x^2 + 11x + 9 = 0 \]
\[ D = 11^2 - 4\cdot2\cdot9 = 121 - 72 = 49 \]
\[ \left(x + 1\right)\left(2x + 9\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x + 9 = 0 \Rightarrow x = -\frac{9}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{4} \]
\[ x = -1\quad\text{or}\quad x = -\frac{9}{2} \]

The parabola y = 2x² + 11x + 9

xy
-5.7511.875
-4.751.875
-3.75-4.125
-2.75-6.125
-1.75-4.125
-0.751.875
0.2511.875

Questions

What are the solutions of 2x² + 11x + 9 = 0?

The solutions of 2x² + 11x + 9 = 0 are x = -1 or x = -9/2.

Does 2x² + 11x + 9 = 0 factor?

Yes: (x+1)(2x+9) = 0.

What is the vertex of y = 2x² + 11x + 9?

(-11/4, -49/8). 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 = 9/2.

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