To solve the quadratic equation [tex]\(4x^2 - 3x + 9 = 2x + 1\)[/tex], we will first bring all terms to one side of the equation to set it to zero. This will give us the quadratic equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
Starting with the original equation:
[tex]\[ 4x^2 - 3x + 9 = 2x + 1 \][/tex]
We subtract [tex]\(2x\)[/tex] and 1 from both sides to get:
[tex]\[ 4x^2 - 3x + 9 - 2x - 1 = 0 \][/tex]
This simplifies to:
[tex]\[ 4x^2 - 5x + 8 = 0 \][/tex]
Now, we have a quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = 8\)[/tex]
Next, we will use the quadratic formula to find the values of [tex]\(x\)[/tex]. The quadratic formula is given by:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
First, we calculate the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-5)^2 - 4(4)(8) \][/tex]
[tex]\[ \Delta = 25 - 128 \][/tex]
[tex]\[ \Delta = -103 \][/tex]
The discriminant [tex]\(\Delta\)[/tex] is [tex]\(-103\)[/tex], which is negative. This tells us there are no real solutions, but there are two complex solutions. We write the solutions using [tex]\(i\)[/tex], the imaginary unit, where [tex]\(i^2 = -1\)[/tex].
The solutions are:
[tex]\[ x = \frac{{-(-5) \pm \sqrt{{-103}}}}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{{5 \pm \sqrt{103} \cdot i}}{8} \][/tex]
Thus, the values of [tex]\(x\)[/tex] are:
[tex]\[ x_1 = \frac{5 + \sqrt{103} \cdot i}{8} \][/tex]
[tex]\[ x_2 = \frac{5 - \sqrt{103} \cdot i}{8} \][/tex]
Therefore, the correct answer is:
[tex]\[ \frac{5 \pm \sqrt{103} i}{8} \][/tex]
