WordGPT

Solve x² - 12x + 9 = 0

The solutions of x² - 12x + 9 = 0 are x = 6 + 3√3 or x = 6 - 3√3, that is x ≈ 11.196152 and x ≈ 0.80384758.

x² +x += 0
x2−12x+9=0⇒x=6+33, 6−33x^2 - 12x + 9 = 0 \Rightarrow x = 6 + 3\sqrt{3},\ 6 - 3\sqrt{3}
Discriminant D = 108 · x ≈ 11.1962, 0.803848

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

Explain it step by step in chatMake a short video of it
Graph of y = x² - 12x + 924681012-30-20-10102030x = 11.196152x = 0.80384758vertexy = x² - 12x + 9
Graph of y = x² - 12x + 9. The curve crosses the x-axis at x = 11.196152 and x = 0.80384758; the vertex is at (6, -27).

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 = 9.

    x2−12x+9=0x^2 - 12x + 9 = 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⋅9=144−36=108D = (-12)^2 - 4\cdot1\cdot9 = 144 - 36 = 108
  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±1082x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{108}}{2}
  5. 5

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

    x=6+33orx=6−33x = 6 + 3\sqrt{3}\quad\text{or}\quad x = 6 - 3\sqrt{3}
  6. 6

    As decimals: x ≈ 11.196152 and x ≈ 0.80384758.

  7. 7

    Check numerically: for x ≈ 11.196152 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 12x + 9 = 0\ \Rightarrow\ x = 6 + 3\sqrt{3},\ 6 - 3\sqrt{3} \]
\[ x^2 - 12x + 9 = 0 \]
\[ D = (-12)^2 - 4\cdot1\cdot9 = 144 - 36 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{108}}{2} \]
\[ x = 6 + 3\sqrt{3}\quad\text{or}\quad x = 6 - 3\sqrt{3} \]

The parabola y = x² - 12x + 9

xy
3-18
4-23
5-26
6-27
7-26
8-23
9-18

Questions

What are the solutions of x² - 12x + 9 = 0?

The solutions of x² - 12x + 9 = 0 are x = 6 + 3√3 or x = 6 - 3√3, that is x ≈ 11.196152 and x ≈ 0.80384758.

Does x² - 12x + 9 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 12x + 9?

(6, -27). 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 = 9.

Similar equations

    Solve x² - 12x + 9 = 0: x = 6 + 3√3 or x = 6 - 3√3 (steps) | WordGPT