WordGPT

Solve x² - 12x + 2 = 0

The solutions of x² - 12x + 2 = 0 are x = 6 + √34 or x = 6 - √34, that is x ≈ 11.830952 and x ≈ 0.16904811.

x² +x += 0
x2−12x+2=0⇒x=6+34, 6−34x^2 - 12x + 2 = 0 \Rightarrow x = 6 + \sqrt{34},\ 6 - \sqrt{34}
Discriminant D = 136 · x ≈ 11.831, 0.169048

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 + 2-22468101214-40-30-20-1010203040x = 11.830952x = 0.16904811vertexy = x² - 12x + 2
Graph of y = x² - 12x + 2. The curve crosses the x-axis at x = 11.830952 and x = 0.16904811; the vertex is at (6, -34).

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 = 2.

    x2−12x+2=0x^2 - 12x + 2 = 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⋅2=144−8=136D = (-12)^2 - 4\cdot1\cdot2 = 144 - 8 = 136
  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±1362x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{136}}{2}
  5. 5

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

    x=6+34orx=6−34x = 6 + \sqrt{34}\quad\text{or}\quad x = 6 - \sqrt{34}
  6. 6

    As decimals: x ≈ 11.830952 and x ≈ 0.16904811.

  7. 7

    Check numerically: for x ≈ 11.830952 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 12x + 2 = 0\ \Rightarrow\ x = 6 + \sqrt{34},\ 6 - \sqrt{34} \]
\[ x^2 - 12x + 2 = 0 \]
\[ D = (-12)^2 - 4\cdot1\cdot2 = 144 - 8 = 136 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{136}}{2} \]
\[ x = 6 + \sqrt{34}\quad\text{or}\quad x = 6 - \sqrt{34} \]

The parabola y = x² - 12x + 2

xy
3-25
4-30
5-33
6-34
7-33
8-30
9-25

Questions

What are the solutions of x² - 12x + 2 = 0?

The solutions of x² - 12x + 2 = 0 are x = 6 + √34 or x = 6 - √34, that is x ≈ 11.830952 and x ≈ 0.16904811.

Does x² - 12x + 2 = 0 factor?

Not over the integers, because the discriminant 136 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 12x + 2?

(6, -34). 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 = 2.

Similar equations

    Solve x² - 12x + 2 = 0: x = 6 + √34 or x = 6 - √34 (steps) | WordGPT