Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To solve the equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex], follow these steps:
### Step 1: Take the Cube Root of Both Sides
The first step to isolate [tex]\(2x - 3\)[/tex] is to take the cube root of both sides of the equation:
[tex]\[ (2x - 3)^3 = \frac{1}{27} \][/tex]
Taking the cube root of both sides, we get:
[tex]\[ 2x - 3 = \sqrt[3]{\frac{1}{27}} \][/tex]
Since [tex]\(\sqrt[3]{\frac{1}{27}} = \frac{1}{3}\)[/tex] (because [tex]\(\left(\frac{1}{3}\right)^3 = \frac{1}{27}\)[/tex]), the equation becomes:
[tex]\[ 2x - 3 = \frac{1}{3} \][/tex]
### Step 2: Solve for [tex]\(x\)[/tex]
Now, isolate [tex]\(x\)[/tex] by first adding 3 to both sides of the equation:
[tex]\[ 2x - 3 + 3 = \frac{1}{3} + 3 \][/tex]
Simplifying the right-hand side:
[tex]\[ 2x = \frac{1}{3} + 3 \][/tex]
Convert 3 to a fraction with the same denominator:
[tex]\[ 2x = \frac{1}{3} + \frac{9}{3} \][/tex]
[tex]\[ 2x = \frac{1 + 9}{3} \][/tex]
[tex]\[ 2x = \frac{10}{3} \][/tex]
Now, divide both sides by 2:
[tex]\[ x = \frac{10}{3} / 2 \][/tex]
[tex]\[ x = \frac{10}{3} \cdot \frac{1}{2} \][/tex]
[tex]\[ x = \frac{10}{6} \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
So, one real solution is:
[tex]\[ x = \frac{5}{3}, \text{ which is approximately } 1.66666666666667 \][/tex]
### Step 3: Consider the Complex Solutions
The equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex] is a cubic equation. A cubic equation generally has three roots—which may include real and complex roots.
Besides the real root [tex]\( x = 1.66666666666667 \)[/tex], there are also two complex roots. These complex roots are:
1. [tex]\( x = 1.41666666666667 - 0.144337567297406i \)[/tex]
2. [tex]\( x = 1.41666666666667 + 0.144337567297406i \)[/tex]
### Summary of Solutions
The three solutions to the equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex] are:
1. [tex]\( x = 1.66666666666667 \)[/tex]
2. [tex]\( x = 1.41666666666667 - 0.144337567297406i \)[/tex]
3. [tex]\( x = 1.41666666666667 + 0.144337567297406i \)[/tex]
### Step 1: Take the Cube Root of Both Sides
The first step to isolate [tex]\(2x - 3\)[/tex] is to take the cube root of both sides of the equation:
[tex]\[ (2x - 3)^3 = \frac{1}{27} \][/tex]
Taking the cube root of both sides, we get:
[tex]\[ 2x - 3 = \sqrt[3]{\frac{1}{27}} \][/tex]
Since [tex]\(\sqrt[3]{\frac{1}{27}} = \frac{1}{3}\)[/tex] (because [tex]\(\left(\frac{1}{3}\right)^3 = \frac{1}{27}\)[/tex]), the equation becomes:
[tex]\[ 2x - 3 = \frac{1}{3} \][/tex]
### Step 2: Solve for [tex]\(x\)[/tex]
Now, isolate [tex]\(x\)[/tex] by first adding 3 to both sides of the equation:
[tex]\[ 2x - 3 + 3 = \frac{1}{3} + 3 \][/tex]
Simplifying the right-hand side:
[tex]\[ 2x = \frac{1}{3} + 3 \][/tex]
Convert 3 to a fraction with the same denominator:
[tex]\[ 2x = \frac{1}{3} + \frac{9}{3} \][/tex]
[tex]\[ 2x = \frac{1 + 9}{3} \][/tex]
[tex]\[ 2x = \frac{10}{3} \][/tex]
Now, divide both sides by 2:
[tex]\[ x = \frac{10}{3} / 2 \][/tex]
[tex]\[ x = \frac{10}{3} \cdot \frac{1}{2} \][/tex]
[tex]\[ x = \frac{10}{6} \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
So, one real solution is:
[tex]\[ x = \frac{5}{3}, \text{ which is approximately } 1.66666666666667 \][/tex]
### Step 3: Consider the Complex Solutions
The equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex] is a cubic equation. A cubic equation generally has three roots—which may include real and complex roots.
Besides the real root [tex]\( x = 1.66666666666667 \)[/tex], there are also two complex roots. These complex roots are:
1. [tex]\( x = 1.41666666666667 - 0.144337567297406i \)[/tex]
2. [tex]\( x = 1.41666666666667 + 0.144337567297406i \)[/tex]
### Summary of Solutions
The three solutions to the equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex] are:
1. [tex]\( x = 1.66666666666667 \)[/tex]
2. [tex]\( x = 1.41666666666667 - 0.144337567297406i \)[/tex]
3. [tex]\( x = 1.41666666666667 + 0.144337567297406i \)[/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.