Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve the equation [tex]\(\frac{2m}{2m + 3} - \frac{2m}{2m - 3} = 1\)[/tex] and determine the number of extraneous solutions, let's go through a step-by-step process.
### Step 1: Combine the Fractions
First, let's combine the fractions on the left-hand side of the equation:
[tex]\[ \frac{2m}{2m + 3} - \frac{2m}{2m - 3} \][/tex]
To combine these fractions, we'll need a common denominator, which is [tex]\((2m + 3)(2m - 3)\)[/tex].
[tex]\[ \frac{2m(2m - 3) - 2m(2m + 3)}{(2m + 3)(2m - 3)} \][/tex]
### Step 2: Simplify the Numerator
Now, let's expand and simplify the numerator:
[tex]\[ 2m(2m - 3) - 2m(2m + 3) = 4m^2 - 6m - (4m^2 + 6m) \][/tex]
Combine like terms in the numerator:
[tex]\[ 4m^2 - 6m - 4m^2 - 6m = -12m \][/tex]
So the combined fraction is:
[tex]\[ \frac{-12m}{(2m + 3)(2m - 3)} \][/tex]
### Step 3: Solve the Equation
The equation now reads:
[tex]\[ \frac{-12m}{(2m + 3)(2m - 3)} = 1 \][/tex]
Multiply both sides by [tex]\((2m + 3)(2m - 3)\)[/tex] to clear the fraction:
[tex]\[ -12m = (2m + 3)(2m - 3) \][/tex]
Recognize that [tex]\((2m + 3)(2m - 3)\)[/tex] is a difference of squares:
[tex]\[ (2m + 3)(2m - 3) = 4m^2 - 9 \][/tex]
So the equation becomes:
[tex]\[ -12m = 4m^2 - 9 \][/tex]
### Step 4: Rearrange and Solve the Quadratic Equation
Rearrange the equation to standard quadratic form:
[tex]\[ 4m^2 - 12m - 9 = 0 \][/tex]
Using the quadratic formula [tex]\(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 4\)[/tex], [tex]\(b = -12\)[/tex], and [tex]\(c = -9\)[/tex]:
[tex]\[ m = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} \][/tex]
Calculate the discriminant:
[tex]\[ \Delta = 144 + 144 = 288 \][/tex]
So:
[tex]\[ m = \frac{12 \pm \sqrt{288}}{8} \][/tex]
Simplify [tex]\(\sqrt{288}\)[/tex]:
[tex]\[ \sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2} \][/tex]
Thus:
[tex]\[ m = \frac{12 \pm 12\sqrt{2}}{8} = \frac{12(1 \pm \sqrt{2})}{8} = \frac{3(1 \pm \sqrt{2})}{2} \][/tex]
The solutions are:
[tex]\[ m = \frac{3(1 + \sqrt{2})}{2} \quad \text{and} \quad m = \frac{3(1 - \sqrt{2})}{2} \][/tex]
### Step 5: Check for Extraneous Solutions
We must ensure that neither solution makes the original denominators zero. We originally stated:
[tex]\[ 2m + 3 = 0 \quad \text{or} \quad 2m - 3 = 0 \][/tex]
1. For [tex]\(m = \frac{3(1 + \sqrt{2})}{2}\)[/tex]:
[tex]\[ 2m = 3(1+\sqrt{2}) \quad \Rightarrow \quad 2m + 3 = 3(1+\sqrt{2}) + 3 \neq 0 \quad \text{and} \quad 2m - 3 = 3(1+\sqrt{2}) - 3 \neq 0 \][/tex]
2. For [tex]\(m = \frac{3(1 - \sqrt{2})}{2}\)[/tex]:
[tex]\[ 2m = 3(1-\sqrt{2}) \quad \Rightarrow \quad 2m + 3 = 3(1-\sqrt{2}) + 3 \neq 0 \quad \text{and} \quad 2m - 3 = 3(1-\sqrt{2}) - 3 \neq 0 \][/tex]
### Conclusion
Since neither solution makes the denominator zero, there are no extraneous solutions.
Final Answer:
[tex]\[ 0 \][/tex]
### Step 1: Combine the Fractions
First, let's combine the fractions on the left-hand side of the equation:
[tex]\[ \frac{2m}{2m + 3} - \frac{2m}{2m - 3} \][/tex]
To combine these fractions, we'll need a common denominator, which is [tex]\((2m + 3)(2m - 3)\)[/tex].
[tex]\[ \frac{2m(2m - 3) - 2m(2m + 3)}{(2m + 3)(2m - 3)} \][/tex]
### Step 2: Simplify the Numerator
Now, let's expand and simplify the numerator:
[tex]\[ 2m(2m - 3) - 2m(2m + 3) = 4m^2 - 6m - (4m^2 + 6m) \][/tex]
Combine like terms in the numerator:
[tex]\[ 4m^2 - 6m - 4m^2 - 6m = -12m \][/tex]
So the combined fraction is:
[tex]\[ \frac{-12m}{(2m + 3)(2m - 3)} \][/tex]
### Step 3: Solve the Equation
The equation now reads:
[tex]\[ \frac{-12m}{(2m + 3)(2m - 3)} = 1 \][/tex]
Multiply both sides by [tex]\((2m + 3)(2m - 3)\)[/tex] to clear the fraction:
[tex]\[ -12m = (2m + 3)(2m - 3) \][/tex]
Recognize that [tex]\((2m + 3)(2m - 3)\)[/tex] is a difference of squares:
[tex]\[ (2m + 3)(2m - 3) = 4m^2 - 9 \][/tex]
So the equation becomes:
[tex]\[ -12m = 4m^2 - 9 \][/tex]
### Step 4: Rearrange and Solve the Quadratic Equation
Rearrange the equation to standard quadratic form:
[tex]\[ 4m^2 - 12m - 9 = 0 \][/tex]
Using the quadratic formula [tex]\(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 4\)[/tex], [tex]\(b = -12\)[/tex], and [tex]\(c = -9\)[/tex]:
[tex]\[ m = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} \][/tex]
Calculate the discriminant:
[tex]\[ \Delta = 144 + 144 = 288 \][/tex]
So:
[tex]\[ m = \frac{12 \pm \sqrt{288}}{8} \][/tex]
Simplify [tex]\(\sqrt{288}\)[/tex]:
[tex]\[ \sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2} \][/tex]
Thus:
[tex]\[ m = \frac{12 \pm 12\sqrt{2}}{8} = \frac{12(1 \pm \sqrt{2})}{8} = \frac{3(1 \pm \sqrt{2})}{2} \][/tex]
The solutions are:
[tex]\[ m = \frac{3(1 + \sqrt{2})}{2} \quad \text{and} \quad m = \frac{3(1 - \sqrt{2})}{2} \][/tex]
### Step 5: Check for Extraneous Solutions
We must ensure that neither solution makes the original denominators zero. We originally stated:
[tex]\[ 2m + 3 = 0 \quad \text{or} \quad 2m - 3 = 0 \][/tex]
1. For [tex]\(m = \frac{3(1 + \sqrt{2})}{2}\)[/tex]:
[tex]\[ 2m = 3(1+\sqrt{2}) \quad \Rightarrow \quad 2m + 3 = 3(1+\sqrt{2}) + 3 \neq 0 \quad \text{and} \quad 2m - 3 = 3(1+\sqrt{2}) - 3 \neq 0 \][/tex]
2. For [tex]\(m = \frac{3(1 - \sqrt{2})}{2}\)[/tex]:
[tex]\[ 2m = 3(1-\sqrt{2}) \quad \Rightarrow \quad 2m + 3 = 3(1-\sqrt{2}) + 3 \neq 0 \quad \text{and} \quad 2m - 3 = 3(1-\sqrt{2}) - 3 \neq 0 \][/tex]
### Conclusion
Since neither solution makes the denominator zero, there are no extraneous solutions.
Final Answer:
[tex]\[ 0 \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
