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Solve -2x² - 5x - 2 = 0

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

x² +x += 0
−2x2−5x−2=0⇒x=−2, −12-2x^2 - 5x - 2 = 0 \Rightarrow x = -2,\ -\frac{1}{2}
Discriminant D = 9 · x ≈ -2, -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 - 5x - 2-4-3-2-112-20-15-10-5x = -2x = -0.5vertexy = -2x² - 5x - 2
Graph of y = -2x² - 5x - 2. The curve crosses the x-axis at x = -2 and x = -0.5; the vertex is at (-1.25, 1.125).

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 = -5, c = -2.

    −2x2−5x−2=0-2x^2 - 5x - 2 = 0
  2. 2

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

    D=(−5)2−4⋅(−2)⋅(−2)=25−16=9D = (-5)^2 - 4\cdot(-2)\cdot(-2) = 25 - 16 = 9
  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+2)(2x+1)=0-\left(x + 2\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+2=0⇒x=−22x+1=0⇒x=−12x + 2 = 0 \Rightarrow x = -2\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±9−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{9}}{-4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -2 back in: a·x² + b·x + c = -2·(4) + (-5)·(-2) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 5x - 2 = 0\ \Rightarrow\ x = -2,\ -\frac{1}{2} \]
\[ -2x^2 - 5x - 2 = 0 \]
\[ D = (-5)^2 - 4\cdot(-2)\cdot(-2) = 25 - 16 = 9 \]
\[ -\left(x + 2\right)\left(2x + 1\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{9}}{-4} \]
\[ x = -2\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = -2x² - 5x - 2

xy
-4.25-16.875
-3.25-6.875
-2.25-0.875
-1.251.125
-0.25-0.875
0.75-6.875
1.75-16.875

Questions

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

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

Does -2x² - 5x - 2 = 0 factor?

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

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

(-5/4, 9/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5/2, product = c/a = 1.

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