Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. $$ \sqrt{t+1}+9=7 $$

Step-by-Step Solution

Answer

No real solution exists; the square root cannot be negative.

Step 1: Isolate the Radical Expression

Begin by isolating the square root expression. You want to get \( \sqrt{t+1} \) by itself on one side of the equation. To do this, subtract 9 from both sides of the original equation: \[ \sqrt{t+1} = 7 - 9 \] This simplifies to: \[ \sqrt{t+1} = -2 \]

Step 2: Analyze the Equation

Notice that you now have \( \sqrt{t+1} = -2 \). The square root of any real number is always non-negative. Hence, a square root cannot equal a negative number like -2. This suggests that there might not be any solution.

Step 3: Conclude About the Solution

Since the square root of a real number cannot be negative, the equation \( \sqrt{t+1} = -2 \) has no valid real solutions. Therefore, no value of \( t \) will satisfy the original equation once simplified.

Key Concepts

Extraneous Solutions

When solving radical equations, you might encounter solutions that appear correct throughout the solving process but do not work in the original equation. These are known as extraneous solutions. They arise mainly when both sides of an equation are manipulated, such as squaring them. It's crucial to check whether your found solution fits into the original equation or not.

• The process of solving can introduce these non-valid solutions, particularly when dealing with squaring or eliminating roots.
• Always substitute your potential solutions back into the original equation to confirm they are not extraneous.
• In our example, even though potential solutions are analyzed, realizing that the square root cannot equal a negative value makes any solution invalid in this context.

Always remember: checking is key!

Isolating Expressions

The cornerstone of solving radical equations is isolating the radical expression. This provides a clear path to solve the equation.

• In our exercise, the goal was to get \( \sqrt{t+1} \) alone on one side of the equation. This involved basic arithmetic operations such as subtraction.
• After isolating the square root, the next steps usually involve further simplification or operations to solve for the variable.
• A critical insight in this step is that isolating the expression helps in understanding what kind of solutions are possible.

Isolating expressions gives a clearer picture and directs the subsequent operations needed to solve the equation.

Real Numbers

Real numbers include all the numbers that can be found on the number line, encompassing both rational and irrational numbers. In the context of radical equations, real numbers guide our understanding of possible solutions.

• When a square root is involved, as in our exercise, remember square roots of real numbers are non-negative.
• This inherent property of non-negativity is crucial in analyzing the validity of solutions.
• Understanding that a real solution to \( \sqrt{t+1} \) must be non-negative helps to quickly see if a solution makes sense.

Keep in mind these properties when dealing with equations involving radicals and roots!

Solving Equations

Solving equations, especially those with radicals, involves understanding the steps and their implications in the realm of real numbers.

• Begin by simplifying the equation—this often means starting with isolating the key expression, as discussed before.
• Check each step for its mathematical validity, ensuring results make sense within established rules like non-negativity of square roots.
• Even when it seems that a mathematical operation will yield a solution, verify it by substituting back into the original equation.

Each solving attempt is a step closer to understanding the true nature of the equation and its solutions!