To solve the quadratic equation [tex]\( 4x^2 + 3 = 2x + 2 \)[/tex], we first need to rewrite it in standard form:
### Step-by-Step Solution:
1. Isolate all terms on one side of the equation:
[tex]\[
4x^2 + 3 - 2x - 2 = 0
\][/tex]
Combine like terms:
[tex]\[
4x^2 - 2x + 1 = 0
\][/tex]
2. Identify the coefficients for the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
a = 4, \quad b = -2, \quad c = 1
\][/tex]
3. Solve using the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute the coefficients into the quadratic formula:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(4)(1)}}{2(4)}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{2 \pm \sqrt{4 - 16}}{8}
\][/tex]
[tex]\[
x = \frac{2 \pm \sqrt{-12}}{8}
\][/tex]
5. Simplify the expression under the square root:
Recognize that [tex]\( \sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i \)[/tex]:
[tex]\[
x = \frac{2 \pm 2\sqrt{3} \cdot i}{8}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{1 \pm \sqrt{3} \cdot i}{4}
\][/tex]
6. Write the solutions clearly:
[tex]\[
x = \frac{1}{4} - \frac{\sqrt{3}}{4} \cdot i \quad \text{and} \quad x = \frac{1}{4} + \frac{\sqrt{3}}{4} \cdot i
\][/tex]
### Conclusion:
Review the options:
- A. [tex]\( e = \frac{1}{8} \)[/tex] – Incorrect
- B. [tex]\( a = -2 \pm \sqrt{3} \)[/tex] – Incorrect
- C. [tex]\( a = 2 \)[/tex] – Incorrect
- D. [tex]\( z = \frac{1 \# \sqrt{3}}{2} \)[/tex] – Incorrect format and expression
None of the given options directly match the solutions we derived. However, our correct answers are:
[tex]\[
\frac{1}{4} - \frac{\sqrt{3}}{4} \cdot i \quad \text{and} \quad \frac{1}{4} + \frac{\sqrt{3}}{4} \cdot i
\][/tex]
Therefore, the correct solutions to the quadratic equation are:
[tex]\[
\frac{1}{4} - \frac{\sqrt{3}}{4} \cdot i \quad \text{and} \quad \frac{1}{4} + \frac{\sqrt{3}}{4} \cdot i
\][/tex]
