91Ó°ÊÓ

In the given equation, we start with an expression involving a quadratic term:
\(10 x^{2} = 80\)
The first goal is to isolate the quadratic term, which in this case is \( x^{2} \). To do this, divide both sides of the equation by 10. This step simplifies the equation and makes it easier to handle.
So, let's divide:
\( \frac{10 x^{2}}{10} = \frac{80}{10} \)
When we simplify both sides, we get:
\( x^{2} = 8 \)
Now, the quadratic term is isolated, meaning it's by itself on one side of the equation. Isolating the term this way makes the next steps more straightforward.

Once you have isolated the quadratic term, the next step is to solve for \( x \) by taking the square root of both sides of the equation.
From our isolated equation:
\( x^{2} = 8 \)
We need to find the value of \( x \) that satisfies this equation. By taking the square root of both sides, we get:
\( x = \pm \sqrt{8} \)
It's important to remember that when you take the square root of a number, you consider both the positive and negative roots. This is because both a positive number squared and a negative number squared can result in the original number.
Simplifying further, \( \sqrt{8} \) can be broken down into a simpler form:
\( 8 = 4 \cdot 2 \), therefore,
\( \sqrt{8} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \)
So the solution to the equation is:
\( x = \pm 2 \sqrt{2} \)

After taking the square root of both sides of the equation, you get two solutions. These are referred to as the positive and negative roots.
In our case, when we took the square root of 8, we considered both:
\( x = \sqrt{8} \) and \( x = -\sqrt{8} \)
This is written together as:
\( x = \pm 2 \sqrt{2} \)
This convention acknowledges that either the positive or the negative root could be a valid solution to the quadratic equation. In problems involving quadratic equations, always remember to include both the positive and negative roots unless specified otherwise.
To summarize, when solving quadratic equations:

• Isolate the quadratic term.
• Take the square root of both sides.
• Include both the positive and negative roots in your solution.