91Ó°ÊÓ

When solving quadratic equations, one of the key steps is isolating the square term. This means we need to rewrite the equation so that the quadratic term (the term with the variable squared) stands alone on one side of the equation.
For example, in the equation \[(x+1)^2 = -9\], the square term \[(x+1)^2\] is already isolated. This makes it easier to analyze the equation and solve for the variable.

In mathematics, real numbers include all the numbers we usually work with, such as positive numbers, negative numbers, and zero. Real numbers can be rational (like fractions) or irrational (like \(\text{√2}\) or \(\text{π}\)).
Key properties of real numbers:

• They can be represented on the number line.
• Real numbers obey the standard operations: addition, subtraction, multiplication, and division (except division by zero).

In the example \[(x+1)^2 = -9\], we consider whether the equation has solutions within the real numbers. Since there is no real number whose square is negative, this particular equation has no solution in the set of real numbers.

Imaginary numbers are used to extend the set of real numbers. They come into play when dealing with the square roots of negative numbers. The basic imaginary unit is \(i\), defined as the square root of -1, so \(i^2 = -1\).
Using this concept, we can solve equations like \((x+1)^2 = -9\) by considering imaginary numbers. We rewrite the equation as:
\[(x+1)^2 = -9\]
Taking the square root of both sides, we get:
\(x+1 = \text{±}i√9\)
This gives us two solutions:
\(x + 1 = 3i\) or \(x + 1 = -3i\)
So, the solutions in terms of x are:
\(x = -1 + 3i\) and \(x = -1 - 3i\)
Imaginary numbers and their combinations with real numbers (called complex numbers) allow us to solve a broader range of equations.