Question

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}-6=0 $$

Short answer

The real solutions are \( x = \sqrt{6} \) and \( x = -\sqrt{6} \)

Step-by-step solution

- Isolate the quadratic term

Rewrite the equation by moving the constant term to the other side: \[ x^2 - 6 = 0 \]Add 6 to both sides: \[ x^2 = 6 \]

- Apply the square root property

Take the square root of both sides of the equation to solve for x: \[ x = \text{±} \sqrt{6} \] This gives two potential solutions because both positive and negative values satisfy the equation.

Key concepts

Solving Quadratics

Solving quadratic equations is a fundamental concept in Algebra. A quadratic equation is any equation that can be written in the standard form: Ax^2 + Bx + C = 0, where A, B, and C are constants, and A ≠ 0. These equations are called 'quadratic' because they involve the square of an unknown variable x ( x^2 ). Solving these equations involves finding the values of x that make the equation true. Depending on the type of quadratic equation, there are various methods to solve them:

• Factoring
• Completing the Square
• Using the Quadratic Formula
• Square Root Property

In our problem, we use the Square Root Property to find the solutions.

Square Root Property

The Square Root Property is a useful tool for solving quadratic equations of the form: x^2 = k, where k is a non-negative constant (k ≥ 0). The Square Root Property states that if x^2 = k , then: x = ±√k. This means that x can be either the positive or the negative square root of k. Let's apply this to our original problem of x^2 - 6 = 0.
First, isolate the quadratic term: x^2 = 6.
Then, use the Square Root Property: x = ±√6.
So, we have two potential solutions: x = √6 or x = -√6.
It’s important to check if all solutions are real numbers, especially when dealing with negative numbers under the square root. In this case, both solutions are real because 6 is positive.

Algebra

In Algebra, understanding the properties of equations and how to manipulate them is key. When solving x^2 - 6 = 0, we apply fundamental Algebra principles:

• Isolate the quadratic term by moving constants to the other side of the equation
• Apply properties like the Square Root Property
• Check all potential solutions

Let's break it down even further:

To isolate x^2, add 6 to both sides: x^2 - 6 + 6 = 0 + 6, which simplifies to: x^2 = 6.

Then, we use the Square Root Property: x = ±√6, leading to two solutions: x = √6 and x = -√6.

This step-by-step approach helps ensure you don’t miss any solutions and can handle more complex quadratics in the future. Algebra is not just about finding the answer but understanding the process to reach it.