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Using the quadratic formula to solve [tex]4x^2 - 3x + 9 = 2x + 1[/tex], what are the values of [tex]x[/tex]?

A. [tex]\frac{1 \pm \sqrt{159}i}{8}[/tex]
B. [tex]\frac{5 \pm \sqrt{153}i}{8}[/tex]
C. [tex]\frac{5 \pm \sqrt{103}i}{8}[/tex]
D. [tex]\frac{1 \pm \sqrt{153}}{8}[/tex]


Sagot :

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]