Problem 22 - Algebra 2

Question

Solve each equation. $$ \sqrt{4 z+1}=3+\sqrt{4 z-2} $$

Step-by-Step Solution

Answer

No solution.

Step 1: Isolate one square root

First, we need to isolate one of the square roots on one side of the equation. The equation is already structured like this. We have \( \sqrt{4z + 1} = 3 + \sqrt{4z - 2} \). No changes necessary for this step.

Step 2: Eliminate the square roots

To eliminate the square roots, we first square both sides of the equation:\[(\sqrt{4z + 1})^2 = (3 + \sqrt{4z - 2})^2\]This gives us:\[4z + 1 = (3 + \sqrt{4z - 2})^2\] which expands to: \( 9 + 6\sqrt{4z - 2} + 4z - 2 \).

Step 3: Simplify the equation

Let's simplify the equation obtained from squaring:\[4z + 1 = 4z + 7 + 6\sqrt{4z - 2}\]Subtract \(4z + 7\) from both sides:\[1 - 7 = 6\sqrt{4z - 2}\]which simplifies to:\[-6 = 6\sqrt{4z - 2}\] and further to:\[-1 = \sqrt{4z - 2}\] This equation is not possible since a square root cannot equal a negative number.

Key Concepts

Isolating Square Roots

When solving equations that involve square roots, a critical step is to isolate one of the square roots on one side of the equation. This means getting the root term by itself, without any other terms, making it easier to manipulate and eventually solve.
For example, consider the equation with two square roots:

• \( \sqrt{4z + 1} = 3 + \sqrt{4z - 2} \)

This equation is already set up with one square root isolated: \( \sqrt{4z + 1} \) is by itself on one side. If it wasn't isolated already, you would need to rearrange terms to achieve this set-up. Isolating a square root simplifies the process of solving the equation by neatly dividing complex parts, thus allowing for a systematic approach to the solution.

Squaring Both Sides

Once you have successfully isolated a square root, the next step is often to square both sides of the equation. This effectively removes the square root, making the equation easier to manage. However, this step requires careful attention to detail to ensure accurate expansion and simplification.
In the given example, once the equation was isolated, we squared both sides:

• The original squared: \( (\sqrt{4z + 1})^2 = (3 + \sqrt{4z - 2})^2 \)

Squaring the left side simply evaluates to \( 4z + 1 \). On the right side, a binomial expansion is required:

• This results in: \( 9 + 6\sqrt{4z - 2} + 4z - 2 \).

This expansion naturally introduces new terms, and a critical point of squaring is ensuring each part of the expression is correctly distributed, precisely calculated, and then methodically simplified.

Impossible Equations

In some situations, the process of solving on paper can lead us to equations that seem unsolvable or lead to contradictions. These equations are sometimes known as impossible equations because they yield results that violate basic mathematical principles, like a root equating to a negative number.
In the exercise provided, after simplification, we ended up with:

• \(-1 = \sqrt{4z - 2} \)

This result is problematic because a square root inherently represents a non-negative number and cannot equal \(-1\). Such outcomes indicate no real solutions exist for the equation given. Recognizing these 'impossible equations' requires checking results at the end of solving radical equations to validate against known mathematical properties, ensuring all steps are correctly interpreted.