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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 ≈ 10.898979 and x ≈ 1.1010205.

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 ≈ 10.899, 1.10102

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

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Graph of y = x² - 12x + 1224681012-30-20-10102030x = 10.898979x = 1.1010205vertexy = x² - 12x + 12
Graph of y = x² - 12x + 12. The curve crosses the x-axis at x = 10.898979 and x = 1.1010205; 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=(−12)2−4⋅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 ≈ 10.898979 and x ≈ 1.1010205.

  7. 7

    Check numerically: for x ≈ 10.898979 the left side is 0 (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
3-15
4-20
5-23
6-24
7-23
8-20
9-15

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 ≈ 10.898979 and x ≈ 1.1010205.

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