Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-17\) solve each equation and check. $$ (2 x)^{\frac{1}{2}}+3=15 $$

Step-by-Step Solution

Answer

The solution is \(x = 72\). Substitute back to check the correctness.

Step 1: Isolate the square root

Subtract 3 from both sides of the equation to isolate the square root term. \[(2x)^{\frac{1}{2}} + 3 = 15\] Subtracting 3 gives us \[(2x)^{\frac{1}{2}} = 12\]

Step 2: Remove the square root

Square both sides of the equation to remove the square root.\[\left((2x)^{\frac{1}{2}}\right)^2 = 12^2\] This simplifies to \[2x = 144\]

Step 3: Solve for x

Divide both sides by 2 to solve for \(x\).\[2x = 144\]Dividing both sides by 2 gives\[x = 72\]

Step 4: Check the solution

Substitute \(x = 72\) back into the original equation to verify:\[(2x)^{\frac{1}{2}} + 3 = 15\]\[\left(2 \times 72\right)^{\frac{1}{2}} + 3 = 15\]\[(144)^{\frac{1}{2}} + 3 = 15\]\[12 + 3 = 15\]Since the left-hand side equals the right-hand side, \(x = 72\) is correct.

Key Concepts

Isolating Variables

When solving equations, one of the primary steps is isolating the variable you are trying to find. In our example, we started with the equation \[(2x)^{\frac{1}{2}} + 3 = 15\]. Our goal is to have \(x\) by itself on one side of the equation. To do this, we should first address any constants around the variable.

• Look at what operations are performed on the variable.
• Use inverse operations to remove these constants.

In this situation, 3 is added to the square root term. Subtracting 3 from both sides will isolate the square root, allowing us to focus solely on \((2x)^{\frac{1}{2}}\). After subtracting 3, we have \[(2x)^{\frac{1}{2}} = 12\].This sets the stage for the next step – dealing with the square root.

Square Roots

Square roots are a common aspect in equations and can sometimes make solving equations tricky. When you have an equation like \[(2x)^{\frac{1}{2}} = 12\], you'll need to "eliminate" the square root to solve for the variable underneath.
To do this, we can square both sides of the equation because squaring a number and taking the square root are inverse operations:

• Squaring the square root term, \(\left((2x)^{\frac{1}{2}}\right)^2\), results in \(2x\).
• Squaring 12 gives you \(144\).

So, this process changes the equation to \[2x = 144\].Now, the equation is much simpler, and solving for \(x\) is straightforward.

Checking Solutions

Once you find a solution to your equation, it's crucial to check that the solution actually works within the context of the original problem. This step ensures that no errors were made during the simplification and solving process.
For our example, we determined that \(x = 72\). Substitute this value back into the original equation \[(2x)^{\frac{1}{2}} + 3 = 15\]. Begin by plugging in 72 for \(x\):

• Calculate \(2 \times 72\) to get \(144\).
• Find the square root of \(144\), which is \(12\).
• Add 3 to 12, and you should get 15.

If both sides of the equation are equal after this computation, your solution is verified! In this case, the solution checks out perfectly, confirming that \(x = 72\) is indeed correct. Checking your solutions is always a good practice to ensure you're on the right track.