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Understanding how to solve quadratic equations is a fundamental skill in algebra. These equations are recognizable by a standard form: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are coefficients, and \( a \) is not equal to zero. The method you use to solve a quadratic equation depends on its structure and complexity.

When an equation is in the form \( ax^2 = k \) (where \( k \) is a constant), a common method is the square root property, used in the given exercise. This method involves isolating the squared term (\( x^2 \) in this case) and taking the square root of both sides. One must remember, however, that applying the square root gives two solutions: one positive and one negative. This is because squaring either a positive or a negative number yields a positive result. It's important to include both solutions when solving these types of equations.

Isolating the variable in an equation is a critical step in the process of solving any algebraic expression. It involves performing algebraic operations to get the variable of interest by itself on one side of the equation. The goal is to rewrite the equation so that one side is just the variable (or the variable to a power), and the opposite side consists solely of numbers or simpler expressions.

For example, in the exercise above, the first step to solve \( 5x^2 = 45 \) involves dividing both sides by 5 to isolate \( x^2 \). This operation yields \( x^2 = 9 \), a simpler equation that can be tackled using the square root property. A thorough grasp of isolating variables is essential for students not only to solve equations but to understand the relationships between different mathematical elements.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). The square root symbol is \( \sqrt{} \) and is fundamental in solving equations involving squares, like \( x^2 = a \).

However, it's important to note that every positive number has two square roots: a positive one and a negative one. This is due to the fact that both positive and negative numbers squared will result in a positive value. So, when you take the square root in an equation, you must consider both the positive root \( (\sqrt{a}) \) and the negative root \( (-\sqrt{a}) \) to ensure you find all possible solutions to the equation. This concept is clearly demonstrated in the provided exercise, where the equation \( x^2 = 9 \) led to two solutions: \( x = 3 \) and \( x = -3 \) upon applying the square root property.