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Chapter 10: Problem 118

Solve the equation. $$\sqrt{2 x-15}-2=3$$

Short Answer

Expert verified
The solution to the equation \(\sqrt{2x-15} - 2 = 3\) is \(x = 20\).

Step by step solution

01

Isolate the square root

To start, the square root must be isolated on one side of the equation. Adding 2 to both sides of the equation \(\sqrt{2x-15} - 2 = 3\), yields \(\sqrt{2x-15} = 5\)
02

Remove square root

To remove the square root, square both sides of the equation. This results to \(2x-15 = 25\)
03

Find \(x\)

Add 15 to both sides to isolate \(2x\) on one side, resulting to \(2x = 40\). Dividing by 2 on both sides of the equation, gives \(x = 20\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Square Roots
In algebraic equations, dealing with square roots requires isolating these roots before solving the equation. This means getting the square root expression by itself on one side of the equation. For instance, in the equation \( \sqrt{2x-15} - 2 = 3 \), the first step is to add 2 to both sides.
This addition cancels the \(-2\) next to the square root. As a result, the equation becomes \( \sqrt{2x-15} = 5 \).
By isolating the square root, you make the equation simpler and prepared for the next steps.
  • Keep the square root on one side.
  • Move all other terms to the opposite side using basic operations like addition or subtraction.

These simple steps pave the way for easier manipulation of the equation.
Solving Algebraic Equations
Solving algebraic equations involves several important processes. Once a square root is isolated, the next phase is eliminating it to simplify the equation further. In our example, \( \sqrt{2x-15} = 5 \), you eliminate the square root by squaring both sides.
When you square the left side, \((\sqrt{2x-15})^2 = 2x-15\), it simplifies to \(2x-15\). Squaring the right side, \(5^2 = 25\). This leaves you with the equation \(2x-15 = 25\).

Now, to solve for \(x\), follow these steps:
  • First, add 15 to each side, leading to \(2x = 40\).
  • Next, divide both sides by 2, resulting in \(x=20\).

Algebraic solutions require keeping the equation balanced by performing the same operation on both sides.
Simplifying Square Roots
Simplifying square roots is a crucial skill in mathematics, aiding in the clarity and resolution of equations. Although the original problem did not involve simplifying the square root directly, understanding this concept is essential. Simplifying square roots means reducing the expression to its simplest form.
For example, if you've got an expression like \( \sqrt{50} \), you find its factors and rewrite it using pairs. Here, \( \sqrt{50} \) can be broken into \( \sqrt{25 \times 2} \) which simplifies to \( 5 \sqrt{2} \).

Steps for simplifying square roots include:
  • Identify perfect squares under the root.
  • Factor them out and rewrite under the square root sign.
  • Bring any perfect square factors out of the root.

Simplification allows for easier computation and simplifies algebraic manipulations.

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