Answered

Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Get immediate and reliable solutions to your questions from a community of experienced professionals 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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.