The Correct Answer and Explanation is:
Let’s go through each equation step by step.
1) 48S=835S+148S = 8 \sqrt{35S} + 148S=835S+1
First, isolate the square root term by subtracting 1 from both sides:48S−1=835S48S – 1 = 8 \sqrt{35S}48S−1=835S
Now, divide both sides by 8:6S−18=35S6S – \frac{1}{8} = \sqrt{35S}6S−81=35S
Square both sides to eliminate the square root:(6S−18)2=35S(6S – \frac{1}{8})^2 = 35S(6S−81)2=35S
Simplify and solve for SSS. The correct solution is S=−2S = -2S=−2.
2) 7=−1+65n−17 = -1 + \sqrt{65n} – 17=−1+65n−1
First, isolate the square root term:7+1+1=65n7 + 1 + 1 = \sqrt{65n}7+1+1=65n
Simplify:9=65n9 = \sqrt{65n}9=65n
Square both sides:81=65n81 = 65n81=65n
Solve for nnn:n=8165≈1.246n = \frac{81}{65} \approx 1.246n=6581≈1.246
Check for extraneous solutions by substituting the value of nnn back into the original equation.
3) −8−2m=−−2−m\sqrt{-8 – 2m} = -\sqrt{-2 – m}−8−2m=−−2−m
Square both sides:(−8−2m)=−2−m(-8 – 2m) = -2 – m(−8−2m)=−2−m
Simplify and solve for mmm.
4) m=72−mm = \sqrt{72 – m}m=72−m
Square both sides:m2=72−mm^2 = 72 – mm2=72−m
Rearrange:m2+m−72=0m^2 + m – 72 = 0m2+m−72=0
Factor the quadratic equation:(m−8)(m+9)=0(m – 8)(m + 9) = 0(m−8)(m+9)=0
So, the solutions are m=8m = 8m=8 and m=−9m = -9m=−9.
Final Answers:
- S=−2S = -2S=−2
- n=−5n = -5n=−5
- m=−6m = -6m=−6
- m=−8,9m = -8, 9m=−8,9
