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To solve the equation \(4 + \sqrt{5x + 66} = x + 10\), follow these steps:
1. Isolate the square root term:
First, isolate the square root on one side of the equation. We can do that by subtracting 4 from both sides:
[tex]\[ \sqrt{5x + 66} = x + 10 - 4 \][/tex]
Simplifying the right side gives us:
[tex]\[ \sqrt{5x + 66} = x + 6 \][/tex]
2. Square both sides of the equation:
To eliminate the square root, square both sides:
[tex]\[ (\sqrt{5x + 66})^2 = (x + 6)^2 \][/tex]
This simplifies to:
[tex]\[ 5x + 66 = (x + 6)(x + 6) \][/tex]
Expanding the right-hand side, we get:
[tex]\[ 5x + 66 = x^2 + 12x + 36 \][/tex]
3. Rearrange the equation into a standard quadratic form:
Now, move all terms to one side of the equation to set it equal to zero:
[tex]\[ 0 = x^2 + 12x + 36 - 5x - 66 \][/tex]
Simplify this:
[tex]\[ 0 = x^2 + 7x - 30 \][/tex]
4. Solve the quadratic equation:
Factor the quadratic equation:
[tex]\[ x^2 + 7x - 30 = 0 \][/tex]
Looking for two numbers that multiply to -30 and add to 7, we find:
[tex]\[ (x + 10)(x - 3) = 0 \][/tex]
Setting each factor equal to zero gives the potential solutions:
[tex]\[ x + 10 = 0 \implies x = -10 \][/tex]
[tex]\[ x - 3 = 0 \implies x = 3 \][/tex]
5. Check the potential solutions in the original equation:
- For \(x = -10\):
[tex]\[ 4 + \sqrt{5(-10) + 66} = -10 + 10 \][/tex]
[tex]\[ 4 + \sqrt{-50 + 66} = 0 \][/tex]
[tex]\[ 4 + \sqrt{16} = 0 \][/tex]
[tex]\[ 4 + 4 = 0 \quad \text{(False)} \][/tex]
So, \(x = -10\) is not a solution.
- For \(x = 3\):
[tex]\[ 4 + \sqrt{5(3) + 66} = 3 + 10 \][/tex]
[tex]\[ 4 + \sqrt{15 + 66} = 13 \][/tex]
[tex]\[ 4 + \sqrt{81} = 13 \][/tex]
[tex]\[ 4 + 9 = 13 \quad \text{(True)} \][/tex]
So, \(x = 3\) is a valid solution.
Hence, the only solution to the equation \(4 + \sqrt{5x + 66} = x + 10\) is:
[tex]\[ \boxed{x = 3} \][/tex]
1. Isolate the square root term:
First, isolate the square root on one side of the equation. We can do that by subtracting 4 from both sides:
[tex]\[ \sqrt{5x + 66} = x + 10 - 4 \][/tex]
Simplifying the right side gives us:
[tex]\[ \sqrt{5x + 66} = x + 6 \][/tex]
2. Square both sides of the equation:
To eliminate the square root, square both sides:
[tex]\[ (\sqrt{5x + 66})^2 = (x + 6)^2 \][/tex]
This simplifies to:
[tex]\[ 5x + 66 = (x + 6)(x + 6) \][/tex]
Expanding the right-hand side, we get:
[tex]\[ 5x + 66 = x^2 + 12x + 36 \][/tex]
3. Rearrange the equation into a standard quadratic form:
Now, move all terms to one side of the equation to set it equal to zero:
[tex]\[ 0 = x^2 + 12x + 36 - 5x - 66 \][/tex]
Simplify this:
[tex]\[ 0 = x^2 + 7x - 30 \][/tex]
4. Solve the quadratic equation:
Factor the quadratic equation:
[tex]\[ x^2 + 7x - 30 = 0 \][/tex]
Looking for two numbers that multiply to -30 and add to 7, we find:
[tex]\[ (x + 10)(x - 3) = 0 \][/tex]
Setting each factor equal to zero gives the potential solutions:
[tex]\[ x + 10 = 0 \implies x = -10 \][/tex]
[tex]\[ x - 3 = 0 \implies x = 3 \][/tex]
5. Check the potential solutions in the original equation:
- For \(x = -10\):
[tex]\[ 4 + \sqrt{5(-10) + 66} = -10 + 10 \][/tex]
[tex]\[ 4 + \sqrt{-50 + 66} = 0 \][/tex]
[tex]\[ 4 + \sqrt{16} = 0 \][/tex]
[tex]\[ 4 + 4 = 0 \quad \text{(False)} \][/tex]
So, \(x = -10\) is not a solution.
- For \(x = 3\):
[tex]\[ 4 + \sqrt{5(3) + 66} = 3 + 10 \][/tex]
[tex]\[ 4 + \sqrt{15 + 66} = 13 \][/tex]
[tex]\[ 4 + \sqrt{81} = 13 \][/tex]
[tex]\[ 4 + 9 = 13 \quad \text{(True)} \][/tex]
So, \(x = 3\) is a valid solution.
Hence, the only solution to the equation \(4 + \sqrt{5x + 66} = x + 10\) is:
[tex]\[ \boxed{x = 3} \][/tex]
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