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Solve -2x² - 11x - 5 = 0

The solutions of -2x² - 11x - 5 = 0 are x = -5 or x = -1/2.

x² +x += 0
−2x2−11x−5=0⇒x=−5, −12-2x^2 - 11x - 5 = 0 \Rightarrow x = -5,\ -\frac{1}{2}
Discriminant D = 81 · x ≈ -5, -0.5

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

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Graph of y = -2x² - 11x - 5-7-6-5-4-3-2-112-30-20-1010x = -5x = -0.5vertexy = -2x² - 11x - 5
Graph of y = -2x² - 11x - 5. The curve crosses the x-axis at x = -5 and x = -0.5; the vertex is at (-2.75, 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 = -11, c = -5.

    −2x2−11x−5=0-2x^2 - 11x - 5 = 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)⋅(−5)=121−40=81D = (-11)^2 - 4\cdot(-2)\cdot(-5) = 121 - 40 = 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+5)(2x+1)=0-\left(x + 5\right)\left(2x + 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:

    x+5=0⇒x=−52x+1=0⇒x=−12x + 5 = 0 \Rightarrow x = -5\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±81−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{81}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−5orx=−12x = -5\quad\text{or}\quad x = -\frac{1}{2}
  8. 8

    Check by putting x = -5 back in: a·x² + b·x + c = -2·(25) + (-11)·(-5) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 11x - 5 = 0\ \Rightarrow\ x = -5,\ -\frac{1}{2} \]
\[ -2x^2 - 11x - 5 = 0 \]
\[ D = (-11)^2 - 4\cdot(-2)\cdot(-5) = 121 - 40 = 81 \]
\[ -\left(x + 5\right)\left(2x + 1\right) = 0 \]
\[ x + 5 = 0 \Rightarrow x = -5\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{81}}{-4} \]
\[ x = -5\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = -2x² - 11x - 5

xy
-5.75-7.875
-4.752.125
-3.758.125
-2.7510.125
-1.758.125
-0.752.125
0.25-7.875

Questions

What are the solutions of -2x² - 11x - 5 = 0?

The solutions of -2x² - 11x - 5 = 0 are x = -5 or x = -1/2.

Does -2x² - 11x - 5 = 0 factor?

Yes: -1(x+5)(2x+1) = 0.

What is the vertex of y = -2x² - 11x - 5?

(-11/4, 81/8). 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 = 5/2.

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