91Ó°ÊÓ

The square root property is a useful tool in solving quadratic equations. This property states that if you have an equation of the form \((x^2 = a)\), then the solutions for \((x)\) are \((x = \pm \sqrt{a})\). It's important to remember to consider both the positive and negative square roots. For example, in our exercise, we start with the equation \(( (3k + 1)^2 = 18 )\). By taking the square root of both sides, we get \(( \sqrt{(3k + 1)^2} = \pm \sqrt{18})\), simplifying to \(( 3k + 1 = \pm \sqrt{18})\).
Understanding the square root property allows us to break down complex quadratic equations into simpler, more manageable parts. Always remember to simplify the result further if possible.

Simplifying radicals is the process of breaking down a radical expression into its simplest form. A radical expression is simplified further when there are no more perfect square factors in the radicand (the number under the radical sign).
Let's take our example from the exercise: \(( \sqrt{18})\). To simplify \(( \sqrt{18})\), we need to find the largest perfect square factor of 18.

• 18 can be factored into 9 and 2, where 9 is a perfect square.
• So, \(( \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2})\).

It's essential to simplify radicals because it makes the equations easier to handle and the solutions more precise. Always look for the largest perfect square factor to ensure your radical is fully simplified.

Isolating variables is a fundamental step in solving equations. This means rearranging the equation so that the variable you are solving for is by itself on one side of the equation.
In our exercise, after applying the square root property, we reach \((3k + 1 = \pm 3 \sqrt{2})\). To isolate \((k)\), we first subtract 1 from both sides: \(( 3k = -1 \pm 3 \sqrt{2})\).

• Next, we divide each term by 3 to solve for \((k)\):
• \(( k = \frac{-1 + 3 \sqrt{2}}{3})\) or \(( k = \frac{-1 - 3 \sqrt{2}}{3})\).

Isolating the variable makes it straightforward to identify the solutions. Breaking down each step and carefully performing each operation helps ensure accuracy and understanding.