Question

Solve the equation. Check your solution(s). \(x^2-4=-11\)

Short answer

The equation \(x^2-4=-11\) does not have any real solutions, because the number under the square root is negative, which isn't possible in the set of real numbers.

Step-by-step solution

Bring all terms to one side of the equation

To start with, we need to move all the terms to one side of the equation to make it easier to solve for x. We bring '-4' to the other side of the equation through addition operation. Thus we get: \(x^2-4 +4 = -11 +4\), which simplifies to \(x^2 = -7\).

Square root both sides

Now, we square root both sides of the equation so as to find the value of x. That gives us: \( \sqrt{x^2} = \sqrt{-7}\). However, note that we don't have real roots for negative numbers. Therefore, the equation has no real solutions.

Key concepts

Algebraic Operations

Algebraic operations form the foundation of solving equations in algebra. They include addition, subtraction, multiplication, division, and the extraction of roots. These operations allow us to manipulate equations into simpler forms, thereby making it possible to solve for unknown variables. In the context of quadratic equations, algebraic operations are especially handy.

For example, to solve the equation like the one given, which is a quadratic equation of the form x² - 4 = -11, our first step involves using addition to isolate the x² term. This means adding 4 to both sides of the equation, which results in x² = -7. This step is crucial as it simplifies the equation, setting the stage for further operations that will help uncover the value of x.

Square Roots

The square root operation is a critical concept when solving quadratic equations. It essentially reverses the action of squaring a number. In other words, if a number y is squared to give y², then the square root of y² is y.

For the equation x² = -7, taking the square root would typically help us find the value of x. However, we encounter a complication since the square root of a negative number isn’t defined within the set of real numbers. This is where the story takes a turn because we have to introduce the concept of imaginary numbers to find the square root of -7. The idea that a square root could result in a non-real number was a significant development in mathematics, paving the way for complex numbers and broader solutions to polynomial equations.

Imaginary Numbers

When we calculate square roots of negative numbers, we enter the realm of imaginary numbers. An imaginary number is defined as the square root of a negative one, and it’s denoted by i. In the case of the equation x² = -7, we are looking for a number, which when squared, gives us -7.

This number can be expressed in terms of imaginary numbers as x = \(±\sqrt{7}i\). The ± symbol indicates that there are two solutions, \sqrt{7}i and -\sqrt{7}i, because both when squared will return our original equation of x² = -7.

Imaginary numbers, and by extension complex numbers, have a wide range of applications in fields such as electrical engineering, signal processing, and quantum physics. They are vital for solving equations that have no real roots, enabling us to find solutions that would otherwise be out of reach.