To solve the quadratic equation [tex]\(4x^2 - 3x + 9 = 2x + 1\)[/tex] using the quadratic formula, follow these detailed steps:
### Step 1: Simplify and Rearrange the Equation
Firstly, the given equation:
[tex]\[4x^2 - 3x + 9 = 2x + 1\][/tex]
needs to be rearranged to standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
Subtract [tex]\(2x + 1\)[/tex] from both sides to get:
[tex]\[4x^2 - 3x + 9 - 2x - 1 = 0\][/tex]
Combine like terms:
[tex]\[4x^2 - 5x + 8 = 0\][/tex]
### Step 2: Identify Coefficients
The standard quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. From the rearranged equation:
[tex]\[4x^2 - 5x + 8 = 0\][/tex]
we identify:
[tex]\[a = 4\][/tex]
[tex]\[b = -5\][/tex]
[tex]\[c = 8\][/tex]
### Step 3: Calculate the Discriminant
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
First, compute the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[\Delta = (-5)^2 - 4 \cdot 4 \cdot 8 = 25 - 128 = -103\][/tex]
### Step 4: Evaluate the Square Root of the Discriminant
Since the discriminant is negative ([tex]\(\Delta = -103\)[/tex]), we have complex roots:
[tex]\[\sqrt{\Delta} = \sqrt{-103} = i\sqrt{103}\][/tex]
### Step 5: Apply the Quadratic Formula
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\sqrt{\Delta}\)[/tex] into the quadratic formula:
[tex]\[x = \frac{-(-5) \pm i\sqrt{103}}{2 \cdot 4} = \frac{5 \pm i\sqrt{103}}{8}\][/tex]
### Conclusion
Thus, the solutions to the equation [tex]\(4x^2 - 3x + 9 = 2x + 1\)[/tex] are:
[tex]\[
x = \frac{5 \pm i \sqrt{103}}{8}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{\frac{5 \pm \sqrt{103} i}{8}}
\][/tex]
