Question

Solve the equation. If there is no solution, state the reason. $$5 x^{2}=-15$$

Short answer

The equation has no solution in the set of real numbers because it involves the square root of a negative number.

Step-by-step solution

Simplify the Equation

The first step is to simplify the equation and isolate \(x\). Start by dividing both sides by 5. This gives: \(x^2 = -3\).

Solve for x

Now, solve for \(x\) by taking the square root of both sides. Remember that when you take the square root of both sides, you must include both the positive and negative roots. This gives: \(x= \pm \sqrt{-3}\).

Consider the Square Root of a Negative Number

Here, you see that you are taking the square root of a negative number. In real numbers, this is not possible because the square of any real number is always positive or zero, never negative. This means the equation has no solution in the set of real numbers.

Key concepts

Simplify the Equation

The process of simplifying an equation is a fundamental step in solving mathematical problems. It involves reducing the equation to its simplest form to make the other steps of finding a solution more straightforward. For the given quadratic equation, \(5 x^{2} = -15\), the first step is to divide both sides by 5 to isolate the \(x^2\) term. Here's a straightforward approach:

• Start with the original equation: \(5 x^{2} = -15\)
• Divide both sides by 5 to simplify: \(\frac{5 x^{2}}{5} = \frac{-15}{5}\)
• After simplifying, the equation becomes \(x^2 = -3\)

At this point, the equation is stripped down to its most basic form, allowing us to address the next step, which is dealing with the square root of negative numbers.

Square Root of Negative Numbers

Encountering a square root of a negative number can be puzzling if you're only familiar with real numbers. Normally, squaring a real number, whether it's positive or negative, yields a positive result. This leads us to an important realization: the square root of a negative number does not exist within the set of real numbers. This concept is crucial when dealing with quadratic equations like \(x^2 = -3\).

In this scenario, the next step would typically involve finding the square root of both sides. However, since we cannot take the square root of a negative number in real numbers, you might wonder how to proceed. This is where complex numbers come into play, specifically the imaginary unit \(i\), which is defined as \(i = \sqrt{-1}\). The expression \(\pm \sqrt{-3}\) can then be understood using this imaginary unit as \(x = \pm i\sqrt{3}\). It's important to note, though, that this solution is not part of the real number system.

No Solution in Real Numbers

When faced with an equation like \(x^2 = -3\), we must accept that not all equations will have a solution in the set of real numbers. This is a key concept in algebra, which acknowledges that the set of real numbers is a field where every positive number has a square root, but the same does not apply to negative numbers. Therefore, when solving quadratic equations, if you find yourself with a negative number under the square root sign, this indicates that the equation has no solution in real numbers.

In such cases, the solution is found within the set of complex numbers, which includes real and imaginary numbers. While this may go beyond the scope of basic algebra, it's important for students to be aware of this distinction. Understanding the limitations within the real number system can help with grasping more advanced mathematical concepts in the future.