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Solve -2x² - 5x - 3 = 0

The solutions of -2x² - 5x - 3 = 0 are x = -3/2 or x = -1.

x² +x += 0
−2x2−5x−3=0⇒x=−32, −1-2x^2 - 5x - 3 = 0 \Rightarrow x = -\frac{3}{2},\ -1
Discriminant D = 1 · x ≈ -1.5, -1

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

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Graph of y = -2x² - 5x - 3-4-3-2-11-16-14-12-10-8-6-4-22x = -1.5x = -1vertexy = -2x² - 5x - 3
Graph of y = -2x² - 5x - 3. The curve crosses the x-axis at x = -1.5 and x = -1; the vertex is at (-1.25, 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 = -5, c = -3.

    −2x2−5x−3=0-2x^2 - 5x - 3 = 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)⋅(−3)=25−24=1D = (-5)^2 - 4\cdot(-2)\cdot(-3) = 25 - 24 = 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:

    −(2x+3)(x+1)=0-\left(2x + 3\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+3=0⇒x=−32x+1=0⇒x=−12x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±1−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−32orx=−1x = -\frac{3}{2}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -3/2 back in: a·x² + b·x + c = -2·(9/4) + (-5)·(-3/2) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 5x - 3 = 0\ \Rightarrow\ x = -\frac{3}{2},\ -1 \]
\[ -2x^2 - 5x - 3 = 0 \]
\[ D = (-5)^2 - 4\cdot(-2)\cdot(-3) = 25 - 24 = 1 \]
\[ -\left(2x + 3\right)\left(x + 1\right) = 0 \]
\[ 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{-4} \]
\[ x = -\frac{3}{2}\quad\text{or}\quad x = -1 \]

The parabola y = -2x² - 5x - 3

xy
-4.25-17.875
-3.25-7.875
-2.25-1.875
-1.250.125
-0.25-1.875
0.75-7.875
1.75-17.875

Questions

What are the solutions of -2x² - 5x - 3 = 0?

The solutions of -2x² - 5x - 3 = 0 are x = -3/2 or x = -1.

Does -2x² - 5x - 3 = 0 factor?

Yes: -1(2x+3)(x+1) = 0.

What is the vertex of y = -2x² - 5x - 3?

(-5/4, 1/8). 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 = 3/2.

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