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Solve x² + 5x + 6 = 0

The solutions of x² + 5x + 6 = 0 are x = -2 or x = -3.

x² +x += 0
x2+5x+6=0⇒x=−2, −3x^2 + 5x + 6 = 0 \Rightarrow x = -2,\ -3
Discriminant D = 1 · x ≈ -2, -3

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

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Graph of y = x² + 5x + 6-5-4-3-2-12468x = -2x = -3vertexy = x² + 5x + 6
Graph of y = x² + 5x + 6. The curve crosses the x-axis at x = -2 and x = -3; the vertex is at (-2.5, -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 = 5, c = 6.

    x2+5x+6=0x^2 + 5x + 6 = 0
  2. 2

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

    D=52−4⋅1⋅6=25−24=1D = 5^2 - 4\cdot1\cdot6 = 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+2)(x+3)=0\left(x + 2\right)\left(x + 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+2=0⇒x=−2x+3=0⇒x=−3x + 2 = 0 \Rightarrow x = -2\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±12x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{1}}{2}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -2 back in: a·x² + b·x + c = 1·(4) + 5·(-2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 5x + 6 = 0\ \Rightarrow\ x = -2,\ -3 \]
\[ x^2 + 5x + 6 = 0 \]
\[ D = 5^2 - 4\cdot1\cdot6 = 25 - 24 = 1 \]
\[ \left(x + 2\right)\left(x + 3\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{1}}{2} \]
\[ x = -2\quad\text{or}\quad x = -3 \]

The parabola y = x² + 5x + 6

xy
-5.58.75
-4.53.75
-3.50.75
-2.5-0.25
-1.50.75
-0.53.75
0.58.75

Questions

What are the solutions of x² + 5x + 6 = 0?

The solutions of x² + 5x + 6 = 0 are x = -2 or x = -3.

Does x² + 5x + 6 = 0 factor?

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

What is the vertex of y = x² + 5x + 6?

(-5/2, -1/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = 6.

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