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Solve 2x² + 5x + 3 = 0

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

x² +x += 0
2x2+5x+3=0⇒x=−1, −322x^2 + 5x + 3 = 0 \Rightarrow x = -1,\ -\frac{3}{2}
Discriminant D = 1 · x ≈ -1, -1.5

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

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Graph of y = 2x² + 5x + 3-4-3-2-11-2246810121416x = -1x = -1.5vertexy = 2x² + 5x + 3
Graph of y = 2x² + 5x + 3. The curve crosses the x-axis at x = -1 and x = -1.5; 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=02x^2 + 5x + 3 = 0
  2. 2

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

    D=52−4⋅2⋅3=25−24=1D = 5^2 - 4\cdot2\cdot3 = 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±14x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{1}}{4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1 back in: a·x² + b·x + c = 2·(1) + 5·(-1) + 3 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 5x + 3 = 0\ \Rightarrow\ x = -1,\ -\frac{3}{2} \]
\[ 2x^2 + 5x + 3 = 0 \]
\[ D = 5^2 - 4\cdot2\cdot3 = 25 - 24 = 1 \]
\[ \left(x + 1\right)\left(2x + 3\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{1}}{4} \]
\[ x = -1\quad\text{or}\quad x = -\frac{3}{2} \]

The parabola y = 2x² + 5x + 3

xy
-4.2517.875
-3.257.875
-2.251.875
-1.25-0.125
-0.251.875
0.757.875
1.7517.875

Questions

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

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

Does 2x² + 5x + 3 = 0 factor?

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

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

(-5/4, -1/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5/2, product = c/a = 3/2.

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