To solve the quadratic equation [tex]\(4x^2 + 3 = 4x + 2\)[/tex], we first need to rearrange it into a standard form of a quadratic equation, which is [tex]\(ax^2 + bx + c = 0\)[/tex].
Step-by-step solution:
1. Start with the original equation:
[tex]\[
4x^2 + 3 = 4x + 2
\][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[
4x^2 + 3 - 4x - 2 = 0
\][/tex]
3. Simplify the equation by combining like terms:
[tex]\[
4x^2 - 4x + 1 = 0
\][/tex]
Now we need to solve this standard form quadratic equation [tex]\(4x^2 - 4x + 1 = 0\)[/tex]. A standard approach for solving a quadratic equation is by using the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
For our quadratic equation [tex]\(4x^2 - 4x + 1 = 0\)[/tex], we identify [tex]\(a = 4\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = 1\)[/tex]. Plugging these values into the quadratic formula, we get:
[tex]\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}
\][/tex]
Simplify inside the square root and the rest of the formula:
[tex]\[
x = \frac{4 \pm \sqrt{16 - 16}}{8}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{0}}{8}
\][/tex]
[tex]\[
x = \frac{4 \pm 0}{8}
\][/tex]
[tex]\[
x = \frac{4}{8}
\][/tex]
[tex]\[
x = \frac{1}{2}
\][/tex]
Thus, the only solution to the quadratic equation [tex]\(4x^2 + 3 = 4x + 2\)[/tex] is:
[tex]\[
\boxed{x = \frac{1}{2}}
\][/tex]
So, the correct answer is:
A. [tex]\(x = \frac{1}{2}\)[/tex]
