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Solve 2x² + 9x = 0

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

x² +x += 0
2x2+9x=0⇒x=0, −922x^2 + 9x = 0 \Rightarrow x = 0,\ -\frac{9}{2}
Discriminant D = 81 · x ≈ 0, -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 + 9x-7-6-5-4-3-2-112-10102030x = 0x = -4.5vertexy = 2x² + 9x
Graph of y = 2x² + 9x. The curve crosses the x-axis at x = 0 and x = -4.5; the vertex is at (-2.25, -10.125).

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 = 9, c = 0.

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

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

    D=92−4⋅2⋅0=81+0=81D = 9^2 - 4\cdot2\cdot0 = 81 + 0 = 81
  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+9)=0\left(x\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=0⇒x=02x+9=0⇒x=−92x = 0 \Rightarrow x = 0\qquad 2x + 9 = 0 \Rightarrow x = -\frac{9}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±814x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{81}}{4}
  7. 7

    Simplify to get the solutions:

    x=0orx=−92x = 0\quad\text{or}\quad x = -\frac{9}{2}
  8. 8

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

LaTeX source of the formulas on this page
\[ 2x^2 + 9x = 0\ \Rightarrow\ x = 0,\ -\frac{9}{2} \]
\[ 2x^2 + 9x = 0 \]
\[ D = 9^2 - 4\cdot2\cdot0 = 81 + 0 = 81 \]
\[ \left(x\right)\left(2x + 9\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 2x + 9 = 0 \Rightarrow x = -\frac{9}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{81}}{4} \]
\[ x = 0\quad\text{or}\quad x = -\frac{9}{2} \]

The parabola y = 2x² + 9x

xy
-5.257.875
-4.25-2.125
-3.25-8.125
-2.25-10.125
-1.25-8.125
-0.25-2.125
0.757.875

Questions

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

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

Does 2x² + 9x = 0 factor?

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

What is the vertex of y = 2x² + 9x?

(-9/4, -81/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = 0.

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