Certainly! Let's solve the quadratic equation step-by-step to find the roots of [tex]\( 4x^2 = 8x - 7 \)[/tex].
### Step-by-Step Solution
#### Step 1: Write the Equation in Standard Form
The given equation is:
[tex]\[ 4x^2 = 8x - 7 \][/tex]
First, we bring all terms to one side of the equation to rewrite it in the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ 4x^2 - 8x + 7 = 0 \][/tex]
#### Step 2: Identify the Coefficients
The standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] allows us to identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 7 \)[/tex]
#### Step 3: Use the Quadratic Formula
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 4 \cdot 7}}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 112}}{8} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{-48}}{8} \][/tex]
#### Step 4: Simplify the Expression
Since we have a negative number inside the square root, this indicates the presence of complex numbers. Simplify the expression further:
[tex]\[ x = \frac{8 \pm \sqrt{-48}}{8} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{48i^2}}{8} \][/tex]
[tex]\[ x = \frac{8 \pm 4\sqrt{3}i}{8} \][/tex]
Divide the numerator by the denominator:
[tex]\[ x = \frac{8}{8} \pm \frac{4\sqrt{3}i}{8} \][/tex]
[tex]\[ x = 1 \pm \frac{\sqrt{3}i}{2} \][/tex]
#### Step 5: Write the Final Roots
The roots of the given quadratic equation are:
[tex]\[ x = 1 + \frac{\sqrt{3}i}{2} \][/tex]
[tex]\[ x = 1 - \frac{\sqrt{3}i}{2} \][/tex]
So, the roots of the equation [tex]\( 4x^2 = 8x - 7 \)[/tex] are [tex]\( x = 1 + \frac{\sqrt{3}i}{2} \)[/tex] and [tex]\( x = 1 - \frac{\sqrt{3}i}{2} \)[/tex].
### Answer
Among the provided choices, the correct one is:
[tex]\[ x = 1 + \frac{\sqrt{3}i}{2}, x = 1 - \frac{\sqrt{3}i}{2} \][/tex]
