Problem 130 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 3-\sqrt{t-3}=\sqrt{t} $$

Step-by-Step Solution

Answer

The solution to the equation is \( t = 4 \).

Step 1: Isolate one square root

To solve the equation, start by isolating one of the square root terms. In this equation, we already have it isolated as given on the left side as \( -\sqrt{t-3} \). However, for convenience, we can bring \( \sqrt{t} \) to one side for subtraction: \( 3 - \sqrt{t} = \sqrt{t-3} \).

Step 2: Eliminate the square root by squaring both sides

Next, eliminate the square root by squaring both sides of the equation. This will allow us to remove one of the square roots: \( (3 - \sqrt{t})^2 = (\sqrt{t-3})^2 \).Simplify both sides:Left side: \( (3 - \sqrt{t})^2 = 9 - 6\sqrt{t} + t \)Right side: \( (\sqrt{t-3})^2 = t-3 \)So the equation becomes \( 9 - 6\sqrt{t} + t = t - 3 \).

Step 3: Simplify and solve for the square root

Now, let's simplify the equation by subtracting \(t\) from both sides: \( 9 - 6\sqrt{t} = -3 \).Then, add 3 to both sides: \( 12 = 6\sqrt{t} \).Divide both sides by 6 to solve for \(\sqrt{t}\): \( 2 = \sqrt{t} \).

Step 4: Solve for t

To solve for \(t\), square both sides of \( 2 = \sqrt{t} \): \( t = 2^2 \).Simplify:\( t = 4 \).

Step 5: Verify the solution

To ensure this solution is correct, substitute \( t = 4 \) back into the original equation: The original equation: \( 3 - \sqrt{t-3} = \sqrt{t} \).Substitute \( t = 4 \): \( 3 - \sqrt{4-3} = \sqrt{4} \), which simplifies to \( 3 - 1 = 2 \), and the right side is also \( 2 \).Since both sides are equal, \( t = 4 \) is indeed the correct solution.

Key Concepts

Isolating Square Roots

Solving square root equations often begins with isolating one of the square roots in the equation. In our problem, we start with the equation \(3 - \sqrt{t-3} = \sqrt{t}\). Our goal is to have a square root alone on one side to simplify our next steps.
You can rearrange terms for simplification reasons. For example, moving terms around to have one square root isolated makes the process of solving more straightforward. This approach helps us eliminate ambiguity and reduces the complexity of the equation.
By transforming the initial equation to \(3 - \sqrt{t} = \sqrt{t-3}\), we have effectively isolated one square root, setting us up for the next step in solving square root equations.

Squaring Both Sides

Once we have isolated a square root, the next step is to eliminate it by squaring both sides of the equation. This removes the square root and allows us to work with a polynomial equation instead.
Given our setup from the previous step, \((3 - \sqrt{t})^2 = (\sqrt{t-3})^2\), we square both sides. This operation yields:

• Left side: \((3 - \sqrt{t})^2 = 9 - 6\sqrt{t} + t\)
• Right side: \((\sqrt{t-3})^2 = t - 3\)

Squaring both sides turns the equation into polynomial terms which are easier to manipulate and solve. Always be cautious: squaring can introduce extraneous solutions, so verification is crucial.

Simplifying Equations

After squaring, we must simplify and solve the resulting equation. Our mission here is to isolate terms with the unknown and solve the equation step by step.
In the example, the equation derived was \(9 - 6\sqrt{t} + t = t - 3\). First, we subtract \(t\) from both sides, simplifying to \(9 - 6\sqrt{t} = -3\). Then, add 3 to each side to get \(12 = 6\sqrt{t}\).
To find \(\sqrt{t}\), divide both sides by 6 which simplifies to \(2 = \sqrt{t}\). This means \(t = 2^2\), thus \(t = 4\). It's essential to verify by substituting \(t = 4\) back into the original equation to ensure our solution satisfies the equation fully. In this instance, it does, which confirms that \(t = 4\) is correct.