91Ó°ÊÓ

Isolating variables is a critical concept in algebra. It involves manipulating an equation so that one variable is alone on one side of the equal sign. This helps simplify the equation and makes it easier to solve.

In the example exercise, the equation is initially given as 7a² = -252.
The first step to isolate the variable term, which is a², is to divide both sides of the equation by 7. This is because 7 is being multiplied by a². By performing the division, we effectively 'cancel out' the 7, leaving us with:
\[\frac{7a^{2}}{7} = \frac{-252}{7} \]
This simplifies to:
\[a^{2} = -36\]
Now, a² is isolated, making it easier to analyze and solve for a.

In algebra, solutions to equations can be either real or imaginary. Real solutions are numbers that can be found on the number line and can be either positive or negative. However, sometimes, especially with quadratic equations, we encounter situations where no real solution exists.

In the isolated equation \[a^{2} = -36\], we are required to find the value of a. Since the square of a real number is always non-negative, a real number squared cannot equal -36. This means there are no real solutions.

To find possible solutions, we use imaginary numbers. The imaginary unit i is defined as \[i = \text{sqrt}(-1)\]. Therefore:
\[a^{2} = -36 \implies a = \text{sqrt}(-36) = \text{sqrt}(-1 \times 36) = \text{sqrt}(-1) \times \text{sqrt}(36) = \text{i} \times 6 = 6\text{i}\]

So, the solutions in this case would be \[-6\text{i}\] and \[6\text{i}\].

Division is a fundamental operation in algebra and is used extensively to solve equations. When we divide both sides of an equation by the same non-zero number, we preserve the equality and simplify the problem.

In the given exercise: 7a² = -252, dividing both sides by 7 is essential to isolate a². Here’s how it works step-by-step:

• Start with the equation 7a² = -252


• Divide each side by 7:
  \[\frac{7a^{2}}{7} = \frac{-252}{7} \]
• This simplifies to:
  \[a^{2} = -36 \]

By isolating a², we’ve simplified the equation, allowing us to determine whether the solutions are real or imaginary. This step-by-step isolation using division is crucial in understanding and solving algebraic equations.