Problem 51 - Algebra 2

Question

Solve each equation. $$ x^{2}-30=-79 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^2 - 30 = -79\) are \(x = 7i\) and \(x = -7i\).

Step 1: Isolate the Square Term

Add 79 to both sides of the equation to isolate the square term, which gives us the equation: \(x^2 = -79 + 30\).

Step 2: Simplify the Equation

Carry out the addition on the right-hand side to simplify the equation to \(x^2 = -49\).

Step 3: Take the Square Root

Since we are looking for the value of x, take the square root of both sides. We need to consider both the positive and negative square roots, so we have \(x = \sqrt{-49}\) or \(x = -\sqrt{-49}\).

Step 4: Solve for x

Square root of a negative number involves the imaginary unit \(i\) where \(i^2 = -1\), thus \(x = \sqrt{49}i\) or \(x = -\sqrt{49}i\), which simplifies to \(x = 7i\) or \(x = -7i\).

Key Concepts

Isolating the Square Term

Solving a quadratic equation often starts with isolating the square term, which is crucial for simplification. It involves rearranging the equation so that the squared variable is on one side of the equation by itself. In the given problem, we start with the equation \(x^2-30=-79\). To isolate \(x^2\), we need to get rid of -30 from the left side. This is done by adding 30 to both sides of the equation, resulting in \(x^2 = -49\).

The importance of isolating the square term lies in setting the stage for further steps that lead to finding the variable's value. Think of it as organizing a room before cleaning it; you clear one area first, making it easier to tackle the rest. Remember, whenever you perform an operation on one side of the equation, you must do the same to the other side to maintain balance.

Taking the Square Root

Once we have isolated the square term, the next step in solving quadratic equations is to 'take the square root' of both sides. This means that we apply the square root operation \(\sqrt{...}\) to undo the squaring of the variable, allowing us to solve for \(x\). In our exercise, after isolating the square term, we have \(x^2 = -49\). When we apply the square root operation, we must consider that every positive number has both a positive and negative square root. Hence, the equation branches into two possible solutions: \(x = \sqrt{-49}\) and \(x = -\sqrt{-49}\).

However, there's a catch! We've encountered a square root of a negative number. This is not possible in the realm of real numbers and instead, we venture into the territory of complex numbers, which introduces us to the concept of imaginary numbers. Nonetheless, taking the square root is a powerful step that moves us further in untangling the variable's value.

Imaginary Numbers

Imaginary numbers may seem perplexing at first, but they are essential for solving equations where we take the square root of a negative number. By definition, an imaginary number is a number that gives a negative result when squared. This is primarily represented by \(i\), where \(i^2 = -1\).

Looking at our exercise, when we take the square root of -49, we're actually taking the square root of -1 times 49, which becomes \(\sqrt{-1} \cdot \sqrt{49}\). Since \(\sqrt{-1}\) is 'i' and \(\sqrt{49}\) is 7, we get \(7i\) as a result. Therefore, we have two complex solutions: \(x = 7i\) and \(x = -7i\), representing both the positive and negative square roots. It's fascinating how imaginary numbers expand the realm of solutions we can have for equations, ensuring that even when faced with the impossible within the real numbers, we always have an answer.