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Solve x² + 3x + 2 = 0

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

x² +x += 0
x2+3x+2=0⇒x=−1, −2x^2 + 3x + 2 = 0 \Rightarrow x = -1,\ -2
Discriminant D = 1 · x ≈ -1, -2

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

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Graph of y = x² + 3x + 2-4-3-2-112468x = -1x = -2vertexy = x² + 3x + 2
Graph of y = x² + 3x + 2. The curve crosses the x-axis at x = -1 and x = -2; the vertex is at (-1.5, -0.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 3, c = 2.

    x2+3x+2=0x^2 + 3x + 2 = 0
  2. 2

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

    D=32−4⋅1⋅2=9−8=1D = 3^2 - 4\cdot1\cdot2 = 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)(x+2)=0\left(x + 1\right)\left(x + 2\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=−1x+2=0⇒x=−2x + 1 = 0 \Rightarrow x = -1\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−3±12x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{1}}{2}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ x^2 + 3x + 2 = 0\ \Rightarrow\ x = -1,\ -2 \]
\[ x^2 + 3x + 2 = 0 \]
\[ D = 3^2 - 4\cdot1\cdot2 = 9 - 8 = 1 \]
\[ \left(x + 1\right)\left(x + 2\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{1}}{2} \]
\[ x = -1\quad\text{or}\quad x = -2 \]

The parabola y = x² + 3x + 2

xy
-4.58.75
-3.53.75
-2.50.75
-1.5-0.25
-0.50.75
0.53.75
1.58.75

Questions

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

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

Does x² + 3x + 2 = 0 factor?

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

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

(-3/2, -1/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 2.

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