91Ó°ÊÓ

When solving quadratic equations, the first step often involves isolating the term that contains the variable. The quadratic equation provided is:

9x^{2} - 25a = 0
To isolate the quadratic term (which in this case is \(9x^2\)), you need to get it on one side of the equation by itself. This can be done by undoing any additions or subtractions involving the other terms. Here, you add \(25a\) to both sides of the equation, resulting in:

9x^{2} = 25a
This step is crucial because it allows us to focus on solving for \(x\), free from other terms. Always remember, to isolate the term means to make it stand alone on one side, making the subsequent steps easier.

Taking the square root is an essential technique when solving for variables in equations involving squares. With the quadratic term isolated, our next step is to solve for \(x\). The simplified equation from our problem looks like this:

x^2 = \frac{25a}{9}
To get rid of the square, we need to take the square root of both sides of the equation. Mathematically, it’s shown as:

\[ x = \text{±}\frac{ \text{√} (25a) }{3} \]
This operation gives us two potential solutions because every positive number has both a positive and a negative square root. Don't forget the ± (plus-minus) sign, which indicates that we acknowledge both positive and negative roots. Simplifying further, we see:

\[ x = \text{±}\frac{ 5 √a }{3} \]
This step is crucial in quadratic equations as it helps break down the problem to find the potential values of \(x\) easily.

Every quadratic equation of the form \(x^2 = k\) where \(k\) is a positive number, will have two solutions: one positive and one negative. These are often referred to as the 'principal square root' (positive) and the 'negative square root'. In the context of our problem, after isolating and simplifying the equation successfully, we end up with:
\[ x = \text{±}\frac{5 \text{√a}}{3} \]Since a quadratic equation represents a parabola, the variable can have two possible solutions: one where \(x\) is positive and one where \(x\) is negative. These solutions come from the nature of squaring — squaring any negative number gives a positive result, just like squaring a positive number does. It is important to consider both solutions as they both accurately solve the given equation. Always double-check your solutions in the context of the original problem. Finally, noting the ± symbol is key.
Understanding that quadratic equations yield two answers is fundamental in mastering these types of mathematical problems.