Problem 48 - Intermediate Algebra

Question

Solve each equation. See Example 7. $$ \sqrt{m+4}=\sqrt{2 m-5} $$

Step-by-Step Solution

Answer

The solution is \( m = 9 \).

Step 1: Eliminate the Square Roots

To eliminate the square roots in the equation \( \sqrt{m+4} = \sqrt{2m-5} \), square both sides of the equation. This gives us: \( m + 4 = 2m - 5 \).

Step 2: Simplify the Equation

Now simplify the equation by getting all the terms involving \( m \) on one side and the constant terms on the other side. Start by subtracting \( m \) from both sides: \( m + 4 - m = 2m - 5 - m \). This simplifies to \( 4 = m - 5 \).

Step 3: Solve for m

Add 5 to both sides of the equation to isolate \( m \): \( 4 + 5 = m - 5 + 5 \). This simplifies to \( 9 = m \).

Step 4: Check Your Solution

Substitute \( m = 9 \) back into the original equation to verify: \( \sqrt{9+4} = \sqrt{2(9)-5} \) gives \( \sqrt{13} = \sqrt{13} \), which is true.

Key Concepts

Square Roots

Square roots are a fundamental concept in algebra that involves finding a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 is 9. In an equation like \( \sqrt{m+4} = \sqrt{2m-5} \), square roots appear on both sides of the equation.
Understanding how to deal with square roots is essential. One effective method to handle equations involving square roots is to eliminate them by squaring both sides. This means that if you have \( \sqrt{x} = y \), by squaring both sides, you turn it into \( x = y^2 \).
It's important to remember that squaring both sides is a powerful tool, but it requires careful handling to ensure no extraneous solutions are introduced. Always check that your calculated solution satisfies the original equation.

Equation Simplification

Simplification is all about making an equation easier to solve. When you start with a complex equation, your goal is to streamline it step by step.
For instance, after squaring both sides of the equation \( \sqrt{m+4} = \sqrt{2m-5} \), it simplifies to \( m+4 = 2m-5 \).
Now, you want to group like terms together. A good strategy is to bring all terms containing \( m \) to one side and constant terms to the other. In the given example, subtracting \( m \) from both sides results in \( 4 = m - 5 \).
Once you have only \( m \) terms on one side, solve for \( m \) by isolating it, leading to a straightforward equation that you can resolve with basic arithmetic.

Equation Verification

Verifying your solution is a crucial final step in solving algebraic equations. Once you've found a potential solution, you must check if it holds true in the original equation.
For the equation \( \sqrt{m+4} = \sqrt{2m-5} \), once you determine \( m = 9 \), you substitute it back to ensure both sides match.
Substitute \( m = 9 \) into the original equation: \( \sqrt{9+4} = \sqrt{2(9)-5} \). This simplifies to \( \sqrt{13} = \sqrt{13} \), confirming that the solution is correct.
Verification helps catch any errors that might have occurred during simplification or squaring, ensuring the solution is valid and not an extraneous answer. Always predictably verify, especially when dealing with squared both sides, which can sometimes introduce false solutions.