Solve the equation [tex]$2 \sqrt{b}+5=11-\sqrt{b}$[/tex].
A) [tex]$b=36$[/tex]
B) [tex]$b=2$[/tex]
C) [tex]$b=4$[/tex]
D) [tex]$b=12$[/tex]
Answer (2)
Add b to both sides: 3 b + 5 = 11 .
Subtract 5 from both sides: 3 b = 6 .
Divide by 3: b = 2 .
Square both sides: b = 4 .
Explanation
Understanding the Problem We are given the equation 2 b + 5 = 11 − b and we want to solve for b .
Isolating the Square Root Term First, we want to isolate the terms with b on one side of the equation and the constant terms on the other side. To do this, we add b to both sides of the equation: 2 b + b + 5 = 11 − b + b 3 b + 5 = 11 Next, we subtract 5 from both sides of the equation: 3 b + 5 − 5 = 11 − 5 3 b = 6
Simplifying the Equation Now, we divide both sides of the equation by 3: 3 3 b = 3 6 b = 2
Solving for b Finally, we square both sides of the equation to solve for b :
( b ) 2 = 2 2 b = 4
Final Answer Therefore, the solution to the equation is b = 4 .
Examples
Imagine you're designing a square garden and need to determine the length of each side. If the area of the garden relates to the side length through a square root equation, solving such an equation helps you find the exact side length needed to achieve the desired area. This principle extends to various scenarios, such as calculating dimensions in construction, determining flow rates in pipes, or even optimizing financial investments where growth is modeled using square roots. Understanding how to manipulate and solve these equations provides a practical tool for precise planning and execution in real-world applications.
The solution to the equation 2 b + 5 = 11 − b is b = 4 , which corresponds to option C). This is found by isolating the square root, simplifying, and squaring both sides of the equation.
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