Problem 37 - Algebra A Combined Function

Question

Solve. $$ \sqrt{x+4}=\sqrt{2 x-5} $$

Step-by-Step Solution

Answer

The solution is \( x = 9 \).

Step 1: Understand the Problem

We have the equation \( \sqrt{x+4} = \sqrt{2x-5} \). Our task is to find the value of \( x \) that satisfies this equation. Both sides involve square roots, so we'll work to isolate \( x \).

Step 2: Square Both Sides

To eliminate the square roots, square both sides of the equation. This gives us:\[(\sqrt{x+4})^2 = (\sqrt{2x-5})^2\]which simplifies to:\[ x + 4 = 2x - 5\]

Step 3: Solve the Linear Equation

Rearrange the equation \( x + 4 = 2x - 5 \) to solve for \( x \). Move \( x \) to one side and constants to the other:\[4 + 5 = 2x - x\]This simplifies to:\[9 = x\]

Step 4: Verify the Solution

Substitute \( x = 9 \) back into the original equation to verify:- \( \sqrt{9+4} = \sqrt{13} \)- \( \sqrt{2(9)-5} = \sqrt{13} \)Both sides are equal, confirming that \( x = 9 \) is a valid solution.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations and involves re-arranging the equation to make it easier to work with. In the exercise given, the first step to solving the equation \( \sqrt{x+4} = \sqrt{2x-5} \) is to apply algebraic manipulation by eliminating the square roots.

To do this, you square both sides of the equation. Squaring is helpful because it removes the radicals, turning the problem into a simpler linear equation. When you square \( \sqrt{x+4} \), you get \( x+4 \). Similarly, squaring \( \sqrt{2x-5} \) gives \( 2x-5 \). This leaves us with the equation \( x + 4 = 2x - 5 \).

By using this technique, you transform a square root equation into a format that is much simpler to solve. Re-arranging and simplifying equations is a handy skill, not just for solving square root equations, but also for tackling a wide array of algebraic expressions.

Linear Equations

Once the equation has been manipulated into a simpler form, we encounter a linear equation. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

In this exercise, the linear equation we derived is \( x + 4 = 2x - 5 \). Solving it involves isolating the variable \( x \) on one side. You can do this by subtracting \( x \) from both sides, which results in \( 4 = x - 5 \).

Next, add 5 to both sides to isolate \( x \), obtaining \( 9 = x \). This means that \( x = 9 \) is the potential solution to the equation. Solving linear equations requires understanding how to balance both sides of an equation through addition, subtraction, multiplication, or division.

Verification of Solutions

Verification is an important step in solving equations as it ensures that the solution is correct. For the equation \( \sqrt{x+4} = \sqrt{2x-5} \), once we found \( x = 9 \), the next step is to check whether this value satisfies the original equation.

To verify, substitute \( x = 9 \) back into the original equation. Calculating gives us \( \sqrt{9+4} = \sqrt{13} \) on the left side and \( \sqrt{2(9)-5} = \sqrt{13} \) on the right side. Both sides are equal, confirming that our solution is indeed correct.

Verification acts as a double-check, assuring us that no mistakes were made during manipulation or simplification. It's a critical part of the process, especially when dealing with more complex equations or when performing operations such as squaring, which might introduce extraneous solutions.