Problem 22 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ \sqrt{x}-15=0 $$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{x}-15=0 \) is \( x = 225 \).

Step 1: Isolate the square root term

The goal is to isolate the square root term, which in this case is \( \sqrt{x} \). This can be done by adding 15 to both sides of the equation which gives \( \sqrt{x} = 15 \).

Step 2: Square both sides

To eliminate the square root, square both sides of the equation. So \((\sqrt{x})^2 = 15^2 \) which simplifies to \( x = 225 \) .

Step 3: Check the solution

To check if we have the right solution, substitute \( x = 225 \) back into the original equation. \( \sqrt{225}-15 = 0 \). And that's true since the square root of 225 is 15. So the solution is correct.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in mathematics where we find the value of one or more unknowns in an equation. At its core, an equation is a statement that two things are equal. In the context of algebraic equations, solving them often involves uncovering the value of a variable, like in the equation \( \sqrt{x}-15=0 \). When solving equations, several straightforward steps can guide you:

• Identify the variable: Recognize which letter represents the unknown value you need to find.
• Isolate the variable: Rearrange the equation so the variable is on one side of the equation by itself.
• Perform operations: Apply mathematical operations to simplify the equation. This could mean addition, subtraction, multiplication, or division.
• Check your solution: Substitute your found value back into the original equation to verify its correctness.

Approaching equations methodically ensures you systematically find the unknowns while maintaining balance and accuracy.

Square Roots

Square roots are mathematical symbols that represent a value which, when multiplied by itself, gives the original number. The square root of a number \( x \) is denoted as \( \sqrt{x} \).For instance, \( \sqrt{225} = 15 \) because \( 15 \times 15 = 225 \). Understanding square roots is crucial when dealing with equations like \( \sqrt{x} = 15 \). To handle square roots when solving equations:

• Recognize the square root in the equation, aiming to "undo" it to find the variable.
• Isolate the square root term first before attempting to eliminate it, ensuring a clean operation afterward.
• Remember that solving square roots can also imply two potential solutions: a positive and a negative, but in the case of principal square roots, we usually take the positive.

Square roots can initially look intimidating, but as you practice, recognizing and manipulating them becomes second nature.

Algebraic Manipulation

Algebraic manipulation involves adjusting and rearranging equations to solve variables. It requires a good understanding of mathematical operations and rules.When you encounter an equation, like \( \sqrt{x} - 15 = 0 \), algebraic manipulation helps simplify and solve it through clear steps:

• Addition/Subtraction: Use these operations to move numbers across the equal sign and isolate terms. For instance, adding 15 to both sides isolates \( \sqrt{x} \).
• Exponentiation: To eliminate square roots, square the term, as done when going from \( \sqrt{x} = 15 \) to \( x = 225 \).
• Balancing: Always maintain balance in an equation; whatever you do to one side, do to the other.

Effective algebraic manipulation means thinking logically about each step, ensuring each operation brings you closer to the solution. Practice is key, revealing pathways to solutions faster each time.