Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To solve the equation
[tex]\[ \sqrt{6 x - 38} = \sqrt{2 x - 9} - \sqrt{4 x - 25} \][/tex]
we need to follow these steps:
1. Isolate one of the square roots:
Given the equation already has isolated square roots on each side, we start manipulating directly.
2. Square both sides:
Squaring both sides helps to remove the square roots:
[tex]\[ (\sqrt{6 x - 38})^2 = \left(\sqrt{2 x - 9} - \sqrt{4 x - 25}\right)^2 \][/tex]
Simplifying this gives:
[tex]\[ 6 x - 38 = (\sqrt{2 x - 9})^2 + (\sqrt{4 x - 25})^2 - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Which simplifies to:
[tex]\[ 6 x - 38 = (2 x - 9) + (4 x - 25) - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Combining like terms:
[tex]\[ 6 x - 38 = 6 x - 34 - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
3. Isolate the remaining square root term:
We need to isolate the term containing the square root:
[tex]\[ 6 x - 38 - 6 x + 34 = - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Simplifying further:
[tex]\[ -4 = - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Dividing by -2 on both sides:
[tex]\[ 2 = \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
4. Square both sides again:
Squaring both sides to remove the square root gives:
[tex]\[ 4 = (2 x - 9)(4 x - 25) \][/tex]
Expanding and simplifying:
[tex]\[ 4 = 8 x^2 - 50 x - 36 x + 225 \][/tex]
Combining like terms:
[tex]\[ 4 = 8 x^2 - 86 x + 225 \][/tex]
Bringing all terms to one side to set the equation to zero:
[tex]\[ 8 x^2 - 86 x + 221 = 0 \][/tex]
5. Solve the quadratic equation:
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 8\)[/tex], [tex]\(b = -86\)[/tex], and [tex]\(c = 221\)[/tex]:
[tex]\[ x = \frac{86 \pm \sqrt{(-86)^2 - 4 \cdot 8 \cdot 221}}{2 \cdot 8} \][/tex]
Simplifying inside the square root and the rest of the formula, we find:
[tex]\[ x = \frac{86 \pm \sqrt{7396 - 7072}}{16} \][/tex]
[tex]\[ x = \frac{86 \pm \sqrt{324}}{16} \][/tex]
[tex]\[ x = \frac{86 \pm 18}{16} \][/tex]
So, we get:
[tex]\[ x = \frac{104}{16} = 6.5 \quad \text{or} \quad x = \frac{68}{16} = 4.25 \][/tex]
6. Verify solutions:
We substitute back into the original equation to verify if [tex]\(x = 6.5\)[/tex] is valid.
Since
[tex]\[ \sqrt{6(6.5) - 38} = \sqrt{2(6.5) - 9} - \sqrt{4(6.5) - 25} \][/tex]
giving
[tex]\[ \sqrt{39 - 38} = \sqrt{13 - 9} - \sqrt{26-25} \][/tex]
which simplifies to:
[tex]\[ \sqrt{1} = \sqrt{4} - \sqrt{1} \][/tex]
[tex]\[ 1 = 2 - 1 = 1 \][/tex]
Thus, we confirm [tex]\(x = 6.5\)[/tex] is a solution.
So, the solution set is [tex]\(\left\{\frac{13}{2}\right\}\)[/tex], which corresponds to option B.
Therefore, the correct answer is:
B. The solution set is the empty set.
[tex]\[ \sqrt{6 x - 38} = \sqrt{2 x - 9} - \sqrt{4 x - 25} \][/tex]
we need to follow these steps:
1. Isolate one of the square roots:
Given the equation already has isolated square roots on each side, we start manipulating directly.
2. Square both sides:
Squaring both sides helps to remove the square roots:
[tex]\[ (\sqrt{6 x - 38})^2 = \left(\sqrt{2 x - 9} - \sqrt{4 x - 25}\right)^2 \][/tex]
Simplifying this gives:
[tex]\[ 6 x - 38 = (\sqrt{2 x - 9})^2 + (\sqrt{4 x - 25})^2 - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Which simplifies to:
[tex]\[ 6 x - 38 = (2 x - 9) + (4 x - 25) - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Combining like terms:
[tex]\[ 6 x - 38 = 6 x - 34 - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
3. Isolate the remaining square root term:
We need to isolate the term containing the square root:
[tex]\[ 6 x - 38 - 6 x + 34 = - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Simplifying further:
[tex]\[ -4 = - 2 \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
Dividing by -2 on both sides:
[tex]\[ 2 = \sqrt{(2 x - 9)(4 x - 25)} \][/tex]
4. Square both sides again:
Squaring both sides to remove the square root gives:
[tex]\[ 4 = (2 x - 9)(4 x - 25) \][/tex]
Expanding and simplifying:
[tex]\[ 4 = 8 x^2 - 50 x - 36 x + 225 \][/tex]
Combining like terms:
[tex]\[ 4 = 8 x^2 - 86 x + 225 \][/tex]
Bringing all terms to one side to set the equation to zero:
[tex]\[ 8 x^2 - 86 x + 221 = 0 \][/tex]
5. Solve the quadratic equation:
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 8\)[/tex], [tex]\(b = -86\)[/tex], and [tex]\(c = 221\)[/tex]:
[tex]\[ x = \frac{86 \pm \sqrt{(-86)^2 - 4 \cdot 8 \cdot 221}}{2 \cdot 8} \][/tex]
Simplifying inside the square root and the rest of the formula, we find:
[tex]\[ x = \frac{86 \pm \sqrt{7396 - 7072}}{16} \][/tex]
[tex]\[ x = \frac{86 \pm \sqrt{324}}{16} \][/tex]
[tex]\[ x = \frac{86 \pm 18}{16} \][/tex]
So, we get:
[tex]\[ x = \frac{104}{16} = 6.5 \quad \text{or} \quad x = \frac{68}{16} = 4.25 \][/tex]
6. Verify solutions:
We substitute back into the original equation to verify if [tex]\(x = 6.5\)[/tex] is valid.
Since
[tex]\[ \sqrt{6(6.5) - 38} = \sqrt{2(6.5) - 9} - \sqrt{4(6.5) - 25} \][/tex]
giving
[tex]\[ \sqrt{39 - 38} = \sqrt{13 - 9} - \sqrt{26-25} \][/tex]
which simplifies to:
[tex]\[ \sqrt{1} = \sqrt{4} - \sqrt{1} \][/tex]
[tex]\[ 1 = 2 - 1 = 1 \][/tex]
Thus, we confirm [tex]\(x = 6.5\)[/tex] is a solution.
So, the solution set is [tex]\(\left\{\frac{13}{2}\right\}\)[/tex], which corresponds to option B.
Therefore, the correct answer is:
B. The solution set is the empty set.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
