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Solve x² + 8x + 12 = 0

The solutions of x² + 8x + 12 = 0 are x = -2 or x = -6.

x² +x += 0
x2+8x+12=0⇒x=−2, −6x^2 + 8x + 12 = 0 \Rightarrow x = -2,\ -6
Discriminant D = 16 · x ≈ -2, -6

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

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Graph of y = x² + 8x + 12-8-7-6-5-4-3-2-1-551015x = -2x = -6vertexy = x² + 8x + 12
Graph of y = x² + 8x + 12. The curve crosses the x-axis at x = -2 and x = -6; the vertex is at (-4, -4).

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 = 8, c = 12.

    x2+8x+12=0x^2 + 8x + 12 = 0
  2. 2

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

    D=82−4⋅1⋅12=64−48=16D = 8^2 - 4\cdot1\cdot12 = 64 - 48 = 16
  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+6)=0\left(x + 2\right)\left(x + 6\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+6=0⇒x=−6x + 2 = 0 \Rightarrow x = -2\qquad x + 6 = 0 \Rightarrow x = -6
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±162x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{2}
  7. 7

    Simplify to get the solutions:

    x=−2orx=−6x = -2\quad\text{or}\quad x = -6
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = 1·(4) + 8·(-2) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 8x + 12 = 0\ \Rightarrow\ x = -2,\ -6 \]
\[ x^2 + 8x + 12 = 0 \]
\[ D = 8^2 - 4\cdot1\cdot12 = 64 - 48 = 16 \]
\[ \left(x + 2\right)\left(x + 6\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x + 6 = 0 \Rightarrow x = -6 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{2} \]
\[ x = -2\quad\text{or}\quad x = -6 \]

The parabola y = x² + 8x + 12

xy
-75
-60
-5-3
-4-4
-3-3
-20
-15

Questions

What are the solutions of x² + 8x + 12 = 0?

The solutions of x² + 8x + 12 = 0 are x = -2 or x = -6.

Does x² + 8x + 12 = 0 factor?

Yes: (x+2)(x+6) = 0.

What is the vertex of y = x² + 8x + 12?

(-4, -4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8, product = c/a = 12.

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