Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

4. Which are the roots of [tex]4x^2 = 8x - 7[/tex]?

A. [tex]x = \frac{2 + i \sqrt{3}}{2}, x = \frac{2 - i \sqrt{3}}{2}[/tex]
B. [tex]x = 1 + \sqrt{3}, x = 1 - \sqrt{3}[/tex]
C. [tex]x = \frac{2 + \sqrt{3}}{2}, x = \frac{2 - \sqrt{3}}{2}[/tex]
D. [tex]x = 1 + i \sqrt{3}, x = 1 - i \sqrt{3}[/tex]

Sagot :

GinnyAnswer
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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.