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Solve x² - 12x + 3 = 0

The solutions of x² - 12x + 3 = 0 are x = 6 + √33 or x = 6 - √33, that is x ≈ 11.744563 and x ≈ 0.25543735.

x² +x += 0
x2−12x+3=0⇒x=6+33, 6−33x^2 - 12x + 3 = 0 \Rightarrow x = 6 + \sqrt{33},\ 6 - \sqrt{33}
Discriminant D = 132 · x ≈ 11.7446, 0.255437

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

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Graph of y = x² - 12x + 3-22468101214-40-30-20-1010203040x = 11.744563x = 0.25543735vertexy = x² - 12x + 3
Graph of y = x² - 12x + 3. The curve crosses the x-axis at x = 11.744563 and x = 0.25543735; the vertex is at (6, -33).

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 = 3.

    x2−12x+3=0x^2 - 12x + 3 = 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⋅3=144−12=132D = (-12)^2 - 4\cdot1\cdot3 = 144 - 12 = 132
  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±1322x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{132}}{2}
  5. 5

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

    x=6+33orx=6−33x = 6 + \sqrt{33}\quad\text{or}\quad x = 6 - \sqrt{33}
  6. 6

    As decimals: x ≈ 11.744563 and x ≈ 0.25543735.

  7. 7

    Check numerically: for x ≈ 11.744563 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 12x + 3 = 0\ \Rightarrow\ x = 6 + \sqrt{33},\ 6 - \sqrt{33} \]
\[ x^2 - 12x + 3 = 0 \]
\[ D = (-12)^2 - 4\cdot1\cdot3 = 144 - 12 = 132 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{132}}{2} \]
\[ x = 6 + \sqrt{33}\quad\text{or}\quad x = 6 - \sqrt{33} \]

The parabola y = x² - 12x + 3

xy
3-24
4-29
5-32
6-33
7-32
8-29
9-24

Questions

What are the solutions of x² - 12x + 3 = 0?

The solutions of x² - 12x + 3 = 0 are x = 6 + √33 or x = 6 - √33, that is x ≈ 11.744563 and x ≈ 0.25543735.

Does x² - 12x + 3 = 0 factor?

Not over the integers, because the discriminant 132 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 12x + 3?

(6, -33). 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 = 3.

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    Solve x² - 12x + 3 = 0: x = 6 + √33 or x = 6 - √33 (steps) | WordGPT