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Solve -2x² - 11x - 9 = 0

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

x² +x += 0
−2x2−11x−9=0⇒x=−92, −1-2x^2 - 11x - 9 = 0 \Rightarrow x = -\frac{9}{2},\ -1
Discriminant D = 49 · x ≈ -4.5, -1

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

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Graph of y = -2x² - 11x - 9-7-6-5-4-3-2-11-30-25-20-15-10-5510x = -4.5x = -1vertexy = -2x² - 11x - 9
Graph of y = -2x² - 11x - 9. The curve crosses the x-axis at x = -4.5 and x = -1; 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=0-2x^2 - 11x - 9 = 0
  2. 2

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

    D=(−11)2−4⋅(−2)⋅(−9)=121−72=49D = (-11)^2 - 4\cdot(-2)\cdot(-9) = 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:

    −(2x+9)(x+1)=0-\left(2x + 9\right)\left(x + 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    2x+9=0⇒x=−92x+1=0⇒x=−12x + 9 = 0 \Rightarrow x = -\frac{9}{2}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = -9/2 back in: a·x² + b·x + c = -2·(81/4) + (-11)·(-9/2) + (-9) = 0 ✓

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

The parabola y = -2x² - 11x - 9

xy
-5.75-11.875
-4.75-1.875
-3.754.125
-2.756.125
-1.754.125
-0.75-1.875
0.25-11.875

Questions

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

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

Does -2x² - 11x - 9 = 0 factor?

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

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

(-11/4, 49/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -11/2, product = c/a = 9/2.

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