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Solve -2x² - 3x - 1 = 0

The solutions of -2x² - 3x - 1 = 0 are x = -1 or x = -1/2.

x² +x += 0
−2x2−3x−1=0⇒x=−1, −12-2x^2 - 3x - 1 = 0 \Rightarrow x = -1,\ -\frac{1}{2}
Discriminant D = 1 · x ≈ -1, -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 - 3x - 1-3-2-112-16-14-12-10-8-6-4-22x = -1x = -0.5vertexy = -2x² - 3x - 1
Graph of y = -2x² - 3x - 1. The curve crosses the x-axis at x = -1 and x = -0.5; the vertex is at (-0.75, 0.125).

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 = -3, c = -1.

    −2x2−3x−1=0-2x^2 - 3x - 1 = 0
  2. 2

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

    D=(−3)2−4⋅(−2)⋅(−1)=9−8=1D = (-3)^2 - 4\cdot(-2)\cdot(-1) = 9 - 8 = 1
  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+1)=0-\left(x + 1\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+1=0⇒x=−12x+1=0⇒x=−12x + 1 = 0 \Rightarrow x = -1\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±1−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{1}}{-4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1 back in: a·x² + b·x + c = -2·(1) + (-3)·(-1) + (-1) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 3x - 1 = 0\ \Rightarrow\ x = -1,\ -\frac{1}{2} \]
\[ -2x^2 - 3x - 1 = 0 \]
\[ D = (-3)^2 - 4\cdot(-2)\cdot(-1) = 9 - 8 = 1 \]
\[ -\left(x + 1\right)\left(2x + 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{1}}{-4} \]
\[ x = -1\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = -2x² - 3x - 1

xy
-3.75-17.875
-2.75-7.875
-1.75-1.875
-0.750.125
0.25-1.875
1.25-7.875
2.25-17.875

Questions

What are the solutions of -2x² - 3x - 1 = 0?

The solutions of -2x² - 3x - 1 = 0 are x = -1 or x = -1/2.

Does -2x² - 3x - 1 = 0 factor?

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

What is the vertex of y = -2x² - 3x - 1?

(-3/4, 1/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3/2, product = c/a = 1/2.

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