Problem 59 - Algebra 1: Concepts and Skills

Question

Which of the following is a solution of \(x=\sqrt{x+20} ?\) F. -5 G. -4 H. 4 J. 5

Step-by-Step Solution

Answer

The solution to the equation \(x = \sqrt{x + 20}\) is J. 5

Step 1: Substitute F

Insert the value of F(-5) into the equation in the place of \(x\): \(-5 \neq \sqrt{-5+20}\). The left-hand side is not equal to the right-hand side, hence option F is not the solution.

Step 2: Substitute G

Insert the value of G(-4) into the equation in the place of \(x\): \(-4 \neq \sqrt{-4+20}\). The left-hand side is not equal to the right-hand side, hence option G is not the solution.

Step 3: Substitute H

Insert the value of H(4) into the equation in the place of \(x\): \(4 = \sqrt{4+20}\). But \(4 \neq \sqrt{24}\), hence option H is not the solution.

Step 4: Substitute J

Insert the value of J(5) into the equation in the place of \(x\): \(5 = \sqrt{5 + 20}\). Because these values are equal (\(5 = 5\)), option J is the solution.

Key Concepts

Substitution Method

When solving radical equations, one common method we use is the substitution method. This process involves testing each potential solution by substituting it into the equation and seeing if it holds true.

• Start by replacing the variable in the equation with each answer choice.
• Perform necessary calculations to determine if both sides of the equation remain equal.
• If the sides are equal, this means the substitution is a correct solution.
• If not, discard that option and move to the next.

For example, in the problem we are examining, the equation is given as:\[ x = \sqrt{x + 20} \] By substituting each of the four answer choices into the equation, we found that only the final option, J, equaled itself when plugged back into the equation, confirming it as the correct solution.

Square Root

In algebra, the square root is a number that, when multiplied by itself, gives the original number. Understanding the square root is crucial when dealing with radical equations.

• The square root symbol is \( \sqrt{} \), and it represents the principal, or non-negative, square root. For example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).
• It is important to remember that the square root of a number cannot be negative.
• In equations like \( x = \sqrt{x + 20} \), solving generally means finding a value for \( x \) that holds the equation true.

Diving into this problem showed us that only when \( x = 5 \) the equation \( x = \sqrt{x + 20} \) makes sense mathematically, because both sides are equal when substituting \( 5 \) for \( x \).

Algebraic Equations

An algebraic equation is a mathematical statement with one or more variables and constants. Solving an algebraic equation means finding the values of the variables that satisfy the equation.

• Equations can include operations like addition, subtraction, multiplication, division, and also roots like square roots.
• The main goal is achieving a point where both sides of the equation have equal values, signifying that a solution is reached.
• For instance, the equation \( x = \sqrt{x + 20} \) is an example of an algebraic equation involving a square root.

When dealing with algebraic equations, especially those that include square roots, it is essential to check each potential solution through substitution. This helps in verifying whether the given potential solutions actually satisfy the equation or not. In our exercise, after testing, it became evident that \( x = 5 \) was the value that made both sides of our equation equal, proving it to be the correct solution.