Question

Solve the equation using square roots.\(x^2-6=30\)

Short answer

The solutions to the equation \(x^2-6=30\) are \(x=6\) and \(x=-6\).

Step-by-step solution

First Isolate \(x^2\)

To solve for \(x\), start by getting \(x^2\) on its own on one side of the equation by adding \(6\) to both sides of the equation \(x^2-6=30\). This will result in \(x^2=36\).

Take the Square Root

Now that \(x^2=36\), the next step is to take the square root of both sides. Don't forget that the square root of a number \(N\) is always two values: positive and negative. This means in our case, \(x= \pm \sqrt{36}\).

Calculate the Final Results

Finally, calculate the square root of \(36\) to get two possible solutions for \(x\), such that \(x=6\) or \(x=-6\). These are the solutions to the original equation.

Key concepts

Square Roots

Square roots are an essential mathematical concept and commonly appear in algebra, especially when solving quadratic equations. Understanding them is crucial. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 36 is 6 because 6 multiplied by 6 equals 36.

When you take the square root of a number, usually denoted by the symbol \(ackslash\sqrt{}\), you need to remember it consists of both a positive and a negative root. This is because both these numbers when squared, give the number we started with. For example, both 6 and -6 squared will result in 36. Hence, when solving quadratic equations like the one we have, each solution involves both the positive and negative square roots.

Isolating the Variable

Isolating the variable means getting the term with the variable by itself on one side of the equation. This is a crucial step in solving any equation and helps to simplify the problem into something more manageable.

For the equation \(x^2 - 6 = 30\), our task was first to isolate \(x^2\). So, we added 6 to both sides of the equation. It's important to perform the same operation on both sides to maintain the equality. This step converted the equation to \(x^2 = 36\). By isolating the variable, it becomes easier to solve for x in the subsequent steps.

Positive and Negative Solutions

When dealing with quadratic equations, especially those involving square roots, it's important to consider both positive and negative solutions. This is because squaring a positive or a negative number results in a positive number.

In the equation where we find \(x^2 = 36\), taking the square root of both sides gives us \(x = \pm \sqrt{36}\). The symbol \pm\ means plus or minus, indicating that there are two possible values for x: one positive and one negative. In our example, it leads to solutions \(x = 6\) and \(x = -6\).

Recognizing that quadratic equations generally have two solutions - a positive and negative - is vital in solving them correctly. This understanding helps avoid missing any potential solutions.