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Solve x² + 12x + 12 = 0

The solutions of x² + 12x + 12 = 0 are x = -6 + 2√6 or x = -6 - 2√6, that is x ≈ -1.1010205 and x ≈ -10.898979.

x² +x += 0
x2+12x+12=0⇒x=−6+26, −6−26x^2 + 12x + 12 = 0 \Rightarrow x = -6 + 2\sqrt{6},\ -6 - 2\sqrt{6}
Discriminant D = 96 · x ≈ -1.10102, -10.899

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

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Graph of y = x² + 12x + 12-12-10-8-6-4-2-30-20-10102030x = -1.1010205x = -10.898979vertexy = x² + 12x + 12
Graph of y = x² + 12x + 12. The curve crosses the x-axis at x = -1.1010205 and x = -10.898979; the vertex is at (-6, -24).

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 = 12, c = 12.

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

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

    D=122−4⋅1⋅12=144−48=96D = 12^2 - 4\cdot1\cdot12 = 144 - 48 = 96
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−12±962x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{96}}{2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−6+26orx=−6−26x = -6 + 2\sqrt{6}\quad\text{or}\quad x = -6 - 2\sqrt{6}
  6. 6

    As decimals: x ≈ -1.1010205 and x ≈ -10.898979.

  7. 7

    Check numerically: for x ≈ -1.1010205 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 12x + 12 = 0\ \Rightarrow\ x = -6 + 2\sqrt{6},\ -6 - 2\sqrt{6} \]
\[ x^2 + 12x + 12 = 0 \]
\[ D = 12^2 - 4\cdot1\cdot12 = 144 - 48 = 96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{96}}{2} \]
\[ x = -6 + 2\sqrt{6}\quad\text{or}\quad x = -6 - 2\sqrt{6} \]

The parabola y = x² + 12x + 12

xy
-9-15
-8-20
-7-23
-6-24
-5-23
-4-20
-3-15

Questions

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

The solutions of x² + 12x + 12 = 0 are x = -6 + 2√6 or x = -6 - 2√6, that is x ≈ -1.1010205 and x ≈ -10.898979.

Does x² + 12x + 12 = 0 factor?

Not over the integers, because the discriminant 96 is not a perfect square. Use the quadratic formula.

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

(-6, -24). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -12, product = c/a = 12.

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    Solve x² + 12x + 12 = 0: x = -6 + 2√6 or x = -6 - 2√6 (steps) | WordGPT