Question

Solve the equation. Check your solution(s). \(x^2+49=0\)

Short answer

The solutions of the equation \(x^2+49=0\) are \(x = 7i\) and \(x = -7i\).

Step-by-step solution

Rearrange the Equation

The goal of this step is to isolate \(x^2\) or find \(x^2\) alone on one side of the equation. One can start by subtracting 49 from both sides of the equation \(x^2 + 49 = 0\) to result in \(x^2 = -49\).

Solve for \(x\)

Because \(x^2 = -49\), to find the value of \(x\), take the square root of both sides. The square root of \(x^2\) is \(x\), and the square root of \(-49\) is \(7i\) and \(-7i\), where \(i\) represents an imaginary number as the square root of a negative number is an imaginary number. Therefore, \(x = 7i, -7i\).

Check Your Solution

To verify that the solutions are correct, substitute \(x = 7i\) and \(x = -7i\) back into the original equation. For \(x = 7i\), the equation becomes \((7i)^2 + 49 = 0\) which simplifies to \(-49 + 49 = 0\), and for \(x = -7i\), it becomes \((-7i)^2 + 49 = 0\) which simplifies to \(-49 + 49 = 0\). Both checks give a true statement, confirming the solutions are correct.

Key concepts

Complex Numbers

When dealing with quadratic equations, we sometimes encounter solutions that don't seem to fit within the realm of real numbers. This is exactly where complex numbers come into play. Imagine you have an equation like the one in our exercise, where you end up with a negative number under a square root. You can't find a real number that, when squared, gives a negative result.

Complex numbers are designed to overcome this limitation. They include a so-called imaginary unit, denoted by '\(i\)', which is defined by the property that \(i^2 = -1\). Therefore, the square root of any negative number can be expressed in terms of \(i\), making it possible to solve equations that have no real solutions. In essence, any complex number can be written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit.

Using Complex Numbers in Equations

In our exercise, when solving for \(x^2 = -49\), we enter the realm of complex numbers by representing the square root of \(-49\) as \(7i\) or \(-7i\). The solutions to our quadratic equation are not just real numbers but complex numbers with zero real part and an imaginary part of \(7i\) or \(-7i\).

Imaginary Numbers

Let's delve a bit deeper into the concept of imaginary numbers. As seen in the exercise, the term 'imaginary' might suggest that these numbers don't exist or are less significant, but that's far from the truth in mathematics. In reality, imaginary numbers are a crucial component of complex numbers and play a pivotal role in various fields, including engineering and physics.

An imaginary number is essentially a real number multiplied by the imaginary unit \(i\). Here's the interesting part: when you square an imaginary number, the result is a real number, but it's negative. This property allows us to work with square roots of negative numbers in a consistent manner. For instance, when we solved for \(x\) in our equation, we found that \(x = \pm\sqrt{-49} = \pm7i\), which is entirely based on the properties of imaginary numbers. It's this flexibility that allows mathematicians and scientists to solve problems that would otherwise be unsolvable.

The Role of Imaginary Numbers in Quadratics

When it comes to quadratic equations, if the discriminant (part under the square root in the quadratic formula) is negative, the solutions involve imaginary numbers. This is an indication that our parabola does not cross the x-axis at any point, and there are no real solutions. Imaginary numbers provide a way to express these 'missing' solutions.

Square Roots

The concept of square roots is core to understanding quadratic equations and navigating complex solutions. A square root of a number \(x\) is a number that, when multiplied by itself, yields \(x\). This might seem straightforward when we're dealing with positive numbers. However, when the original equation yields a negative number under the square root sign, we enter the territory of imaginary numbers, as we've discussed above.

For positive numbers, taking a square root is quite intuitive. For example, the square root of 9 is 3, because \(3^2 = 9\). But with negative numbers, since no real number squared gives a negative result, we express the square root using \(i\), the basic unit of imaginary numbers. That is why in our exercise, the square root of \(-49\) becomes \(7i\), following the logic we've established with imaginary numbers.

Square Roots and Their Properties

A distinctive feature of square roots is the presence of two solutions: one positive and one negative, usually represented as \(\pm\sqrt{x}\). This is also reflected in the nature of quadratic solutions, where we might have two different real solutions, one real solution (in the case of a perfect square), or two imaginary solutions if the value under the square root is negative.