To find the solutions for the quadratic equation [tex]\(4x^2 + 3 = 4x + 2\)[/tex], we need to follow these steps:
1. Rearrange the equation to standard form:
Start with the given equation:
[tex]\[
4x^2 + 3 = 4x + 2
\][/tex]
Subtract [tex]\(4x\)[/tex] and 2 from both sides to arrange it into standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
4x^2 - 4x + 3 - 2 = 0
\][/tex]
Simplify:
[tex]\[
4x^2 - 4x + 1 = 0
\][/tex]
2. Identify the coefficients:
In the standard quadratic equation [tex]\(4x^2 - 4x + 1 = 0\)[/tex], the coefficients are:
[tex]\[
a = 4, \quad b = -4, \quad c = 1
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(D\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
D = (-4)^2 - 4 \cdot 4 \cdot 1 = 16 - 16 = 0
\][/tex]
4. Determine the nature of the roots:
Since the discriminant [tex]\(D = 0\)[/tex], this indicates that the quadratic equation has exactly one real double root.
5. Solve using the quadratic formula:
The quadratic formula [tex]\(x\)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{D}}{2a}
\][/tex]
With [tex]\(D = 0\)[/tex]:
[tex]\[
x = \frac{-b \pm 0}{2a} = \frac{-b}{2a}
\][/tex]
Substitute [tex]\(b = -4\)[/tex] and [tex]\(a = 4\)[/tex]:
[tex]\[
x = \frac{-(-4)}{2 \cdot 4} = \frac{4}{8} = \frac{1}{2}
\][/tex]
Thus, the solution to the quadratic equation [tex]\(4x^2 - 4x + 1 = 0\)[/tex] is [tex]\(x = \frac{1}{2}\)[/tex].
Considering the choices given:
A. [tex]\(x = \frac{1}{2}\)[/tex]
B. [tex]\(x = -2 \pm \sqrt{3}\)[/tex]
C. [tex]\(x = 2\)[/tex]
D. [tex]\(x = \frac{1 \pm \sqrt{3}}{2}\)[/tex]
The correct answer is:
A. [tex]\(x = \frac{1}{2}\)[/tex]
