Problem 49 - Precalculus : Building Concepts and Connections

Question

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt{x+3}+\sqrt{x-5}=4$$

Step-by-Step Solution

Answer

The solution to the radical equation is \(x = 6\).

Step 1: Isolate one of the square roots

Subtract \(\sqrt{x-5}\) from both sides of the equation to isolate \(\sqrt{x+3}\) on one side of the equation. The equation now reads \(\sqrt{x+3} = 4 - \sqrt{x-5}\).

Step 2: Eliminate the square root

Square both sides of the equation in order to remove the radicals. This gives us: \((\sqrt{x+3})^2 = (4 - \sqrt{x-5})^2\). Simplifying this further, we get \(x + 3 = 16 - 8\sqrt{x-5} + x - 5\).

Step 3: Simplify and isolate the other square root

Simplify the equation to isolate the other square root. Subtract x from both sides and add \(8\sqrt{x-5}\) to both sides, leading to: \(8 = 8\sqrt{x-5}\).

Step 4: Solve for x

Divide both sides by 8: \(1 = \sqrt{x-5}\). Now, square both sides to get rid of the square root: \(1 = x - 5\). Finally, add 5 to both sides to find that \(x = 6\).

Step 5: Check the solution

Substitute 6 back into the original equation to check if it is a valid solution: \(\sqrt{6+3} + \sqrt{6-5} = 4\). This simplifies to 3 + 1 = 4, which is indeed true. Therefore, \(x = 6\) is a valid solution.

Key Concepts

Square Roots

A square root is a way to find a number which, when multiplied by itself, gives us the original number. The symbol for square root is \(\sqrt{}\). For example, \(\sqrt{9}\) is 3, because 3 times 3 equals 9. Square roots are often found in equations where the unknown variable is inside the square root symbol. In these kinds of equations, called radical equations, our goal is usually to "remove" the square root so that we can solve for the variable more easily.

To accomplish this in equations, we use a method called squaring both sides. When you square something, you multiply it by itself. Doing this with the square root, such as \((\sqrt{x})^2\), cancels it out because squaring a square root gives just the original inside value \(x\). Remembering this concept will help you tackle problems where the variables are under square roots.

Isolating Variables

In mathematics, isolating a variable means rearranging an equation so that the variable is alone on one side. This is essential for solving equations, as it helps you to find the value of the unknown variable. To isolate a variable within a radical equation:

• Start by getting the square root term alone on one side of the equation. This might involve addition or subtraction of other terms.
• Once isolated, square both sides of the equation to eliminate the square root. Doing so helps us see a clearer path to solve for the variable.

For example, if we start with \(\sqrt{x+3} = 4 - \sqrt{x-5}\), we need to eliminate one square root to solve for \(x\).

Isolation makes solving complex equations simpler because it gradually reduces the number of operations or terms you have to handle. Mastery of this method is crucial for solving any kind of algebraic equation.

Checking Solutions

After solving an equation, especially those involving square roots, it is crucial to verify that your solution is correct. This step, called checking your solution, uncovers any possible errors made during the squaring process.

• Take the value found for \(x\) and substitute it back into the original equation.
• Perform the necessary operations to confirm that both sides of the equation remain equal.

For instance, when you solve the equation \(\sqrt{x+3} + \sqrt{x-5} = 4\) and find that \(x = 6\), substitute 6 back into the original equation to ensure it holds true.

Checking solutions is paramount to tackling extraneous solutions — values that seem correct mathematically but don’t satisfy the original equation. This often happens because squaring both sides can introduce solutions that weren’t part of the original setup.