Question

In Exercises 15–26, solve the equation. Check your solution(s). $$ \sqrt{3 x-3}-\sqrt{x+12}=0 $$

Short answer

The solution to the equation is \(x = 7.5\).

Step-by-step solution

Isolate the square root

The first step is to isolate one of the square roots. By adding \(\sqrt{x+12}\) to both sides we get: \(\sqrt{3 x-3}= \sqrt{x+12}\)

Square both sides

To remove the square root, square both sides. This will give: \(3x - 3 = x + 12\) which simplifies to \(2x = 15\).

Solve for x

To find the value of x, divide both sides by 2. This gives: \(x = 15/2 = 7.5\)

Verify the solution

The last step is to verify the solution by substituting x = 7.5 into the original equation and confirm the left and right sides are equal. When checked, you will see that they both equal \(\sqrt{9}\), so \(x = 7.5\) is a valid solution.

Key concepts

Isolating the Square Root

When tackling a square root equation, the first order of business is to isolate the square root on one side of the equation. This is a crucial step because it allows us to deal with the square root directly and simplifies the problem. Imagine a seesaw, where balance is achieved when each side has a similar weight. In mathematical terms, we want to move every term not under the square root to the other side of the equal sign, just like moving weight to one side of the seesaw. This is done through simple operations such as addition or subtraction.

For example, consider the equation \(\sqrt{3x - 3} - \sqrt{x + 12} = 0\). To isolate \(\sqrt{3x - 3}\), we add \(\sqrt{x + 12}\) to both sides of the equation, resulting in \(\sqrt{3x - 3} = \sqrt{x + 12}\). By doing so, we have successfully 'lifted' the square root onto one side of the seesaw, making our next steps clearer.

Squaring Both Sides of an Equation

Once we have isolated the square root on one side, we can proceed to eliminate it by using the next crucial technique, 'squaring both sides of the equation.' Squaring a number means multiplying the number by itself, and it is an appropriate method here because it is the inverse operation of taking the square root. This action is akin to unlocking a door – it allows us to access the values within the square root.

In the previous example, after isolating the square root, we had \(\sqrt{3x - 3} = \sqrt{x + 12}\). When we square both sides, we are left with \(3x - 3 = x + 12\). This step has removed the square root and allowed us to proceed to an algebraic equation that we can solve using familiar methods. It is imperative, however, to remember to square the entire side of the equation, not just the square root, to maintain the equation's balance.

Radical Equations

Radical equations, which contain radical signs (like square roots), pose their unique challenges and require specific steps for a solution. These equations can sometimes lead to extraneous solutions—answers that fit the transformed equation but do not satisfy the original equation. It's the mathematical equivalent of a mirage, an answer that seems correct but disappears upon closer examination.

Therefore, after solving the equation \(3x - 3 = x + 12\) to find \(x = 7.5\), we must always plug this solution back into the original equation to verify. This process is the final checkpoint to ensure we haven't been fooled by an extraneous solution. When we check \(x = 7.5\) in the original \(\sqrt{3x - 3} - \sqrt{x + 12} = 0\), both sides calculate to \(\sqrt{9}\), which confirms that \(x = 7.5\) is indeed a valid solution. Maintaining vigilance throughout every step is key to mastering radical equations.