91Ó°ÊÓ

Square roots are an essential part of mathematics, particularly when dealing with quadratic equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6, because 6 multiplied by 6 equals 36.
In equations like \( \sqrt{x-4} = 6 \), the goal is often to "undo" the square root. This is done by squaring both sides of the equation. The process essentially reverses applying a square root, allowing us to see more clearly the operations needed to solve for the unknown variable.
To practically remove the square root, you square both sides:

• Squaring \( \sqrt{x-4} \) simplifies it to \( x-4 \).
• Squaring 6 results in 36.

After this step, the equation simply becomes \( x - 4 = 36 \), which is much easier to solve.

Solving an equation means finding the value of the variable that makes the equation true. For equations involving square roots, it's important to first isolate the square root on one side before proceeding with other operations.
In our exercise, the equation was \( \sqrt{x-4} = 6 \). Here, the square root was already isolated, meaning it was the only term on one side of the equation. This made it simpler to move to the next step of squaring both sides.
Remember to always:

• Isolate the square root or any other main operation.
• Ensure all sides of the equation are simplified before attempting to solve for the variable.

By carefully following this process, you'll be able to solve virtually any simple equation involving square roots.

Algebraic manipulation involves using mathematical operations to transform an equation into a simpler form. This is crucial in making complex equations easier to solve.
In the context of our exercise, after isolating the square root and squaring both sides, we used algebraic manipulation to simplify the expression \( x - 4 = 36 \). To solve for \( x \), we "undo" the subtraction by adding 4 to both sides. This tracks as:

• Starting with \( x - 4 = 36 \).
• Add 4 to 36 to get \( 40 \).
• Therefore, \( x = 40 \).

These basic principles of adding, subtracting, multiplying, or dividing terms are key to manipulating equations accurately and solving them efficiently.