91Ó°ÊÓ

When dealing with quadratic equations like \((x-1)^2 = -9\), you often encounter complex solutions. Complex numbers are numbers that have both a real part and an imaginary part. In this problem, the number \(-9\) on the right side leads us to complex solutions because the square root of a negative number involves an imaginary unit, denoted as \(i\). In this context, \(i\) is defined as \(\sqrt{-1}\).When applying the square root property, you take the square root of both sides, resulting in:

• \(x-1 = \pm i\sqrt{9}\)

From this, you derive two separate equations: \(x-1 = 3i\) and \(x-1 = -3i\). Here, \(3i\) and \(-3i\) are your complex solutions for the variable \(x\). They exemplify how imaginary numbers work as solutions for squares of negative numbers.

Isolating expressions is an essential step in solving equations like \((x-1)^2 = -9\). To isolate an expression means to get it by itself on one side of the equation. This makes it easier to solve the equation through simplification.Initially, the expression \((x-1)\) is already squared and isolated on the left side. The task is then simplified as nothing needs to be moved around. If it weren't isolated, you'd need to perform operations such as addition or subtraction to both sides to isolate it.This isolated form \((x-1)^2 = -9\) is ideal for easily applying the square root property. Remember, isolation is crucial because it sets up the equation in a way that allows you to apply further steps effectively.

Solving quadratic equations by using the square root property is a method that can be very efficient when applicable. The square root property states that if \((x-c)^2 = k\), then \(x-c = \pm \sqrt{k}\). In this exercise, you begin with the equation \((x-1)^2 = -9\). By taking the square root of both sides, you effectively remove the square on the left side. This turns into \(x-1 = \pm i\sqrt{9}\), leading to two separate equations:

• \(x-1 = 3i\)
• \(x-1 = -3i\)

The next step is straightforward: solve each equation to find the value of \(x\). For instance, add 1 to both sides of each equation to get \(x = 1 + 3i\) and \(x = 1 - 3i\). This step-by-step approach helps in efficiently solving the equation while ensuring understanding at each level.