$$ 2.12 x^{2}+2 x+1=0 $$ solve for $$ x $$

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We are given the quadratic equation

$$
2.12x^2 + 2x + 1 = 0.
$$

**Step 1. Identify the coefficients.**

Comparing with the standard form

$$
ax^2 + bx + c = 0,
$$

we have:

$$
a = 2.12,\quad b = 2,\quad c = 1.
$$

**Step 2. Write down the quadratic formula.**

The quadratic formula is given by

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$

**Step 3. Substitute the values into the formula.**

Substitute $$a = 2.12$$, $$b = 2$$, and $$c = 1$$:

$$
x = \frac{-2 \pm \sqrt{2^2 - 4(2.12)(1)}}{2(2.12)}.
$$

**Step 4. Calculate the discriminant.**

The discriminant $$\Delta$$ is

$$
\Delta = b^2 - 4ac = 2^2 - 4(2.12)(1) = 4 - 8.48 = -4.48.
$$

Since $$\Delta < 0$$, the solutions will be complex.

**Step 5. Express the solution with the complex term.**

The square root of the negative discriminant can be written as:

$$
\sqrt{-4.48} = i\sqrt{4.48}.
$$

Thus, the solutions become:

$$
x = \frac{-2 \pm i\sqrt{4.48}}{4.24}.
$$

**Step 6. Simplify (optionally using decimal approximations).**

We can simplify further by approximating the square root:

$$
\sqrt{4.48} \approx 2.116.
$$

Then the solutions are approximately:

$$
x \approx \frac{-2 \pm 2.116i}{4.24}.
$$

Dividing numerator and denominator gives:

$$
x \approx -0.4717 \pm 0.5i.
$$

**Final Answer:**

$$
x = \frac{-2 \pm i\sqrt{4.48}}{4.24} \quad \text{or approximately} \quad x \approx -0.4717 \pm 0.5i.
$$
The solutions are $$ x = \frac{-2 \pm i\sqrt{4.48}}{4.24} $$ or approximately $$ x \approx -0.4717 \pm 0.5i $$.

\( 2.12 x^{2}+2 x+1=0 \) solve for \( x \)

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