Solve each equation using the square root property. See Example 2. 27 - x^2 = 0

Verified step by step guidance

1. Start by isolating the term with the variable. In this case, add $x^2$ to both sides of the equation to get $27=x^2$.
2. Next, apply the square root property, which states that if $a^2=b$, then $a=\pm \sqrt{b}$.
3. Take the square root of both sides of the equation: $x=\pm \sqrt{27}$.
4. Simplify the square root if possible. Note that $\sqrt{27}=\sqrt{9\times 3}=\sqrt{9}\times \sqrt{3}=3\sqrt{3}$.
5. Thus, the solutions to the equation are $x=3\sqrt{3}$ and $x=-3\sqrt{3}$.

Key Concepts

Square Root Property

The square root property states that if x^2 = k, then x = ±√k. This property is essential for solving quadratic equations, as it allows us to isolate the variable by taking the square root of both sides. It is particularly useful when the equation can be rearranged into the form of a perfect square.

Rearranging Equations

Rearranging equations involves manipulating the equation to isolate the variable of interest. In the context of the given problem, we first need to move all terms to one side to set the equation to zero, which is a common practice in solving equations. This step is crucial for applying the square root property effectively.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. The equation in the question, 27 - x^2 = 0, can be recognized as a quadratic equation. Understanding the characteristics of quadratic equations helps in identifying appropriate methods for solving them, such as factoring, completing the square, or using the square root property.