Question

Solve the equation. Check your solution(s). \(x^2+7=-47\)

Short answer

The equation has no real number solutions.

Step-by-step solution

Rearrange the equation

Rearrange the equation to move the constant on the left-hand side over to the right-hand side so that it becomes \(x^2=-47-7\). After simplifying, the equation becomes \(x^2=-54\).

Solve the equation

To solve for x, take the square root of both sides of the equation. This gives \(x=\sqrt{-54}\). However, since we can't take the square root of a negative number in the real number system, there are no real number solutions to this equation.

Check the solution

As there is no real number solution, there's no solution to check.

Key concepts

Understanding the Square Root

The square root is a mathematical operation that finds the number which, when multiplied by itself, yields the original number. For example, the square root of 25 is 5, because when you multiply 5 by 5, you get 25. This operation is commonly represented by the radical symbol \(\sqrt{}\). It's important to note that while every positive number has two square roots (one positive and one negative), the square root symbol typically refers to the principal or non-negative root.
In the context of our exercise, we attempted to find the square root of \(-54\). However, negative numbers do not have real square roots. This is because no real number multiplied by itself results in a negative product. This is why the square root of a negative number does not exist within the realm of real numbers and instead lies in the realm of complex numbers.

Real Numbers Explanation

Real numbers are a broad category of numbers that include both rational numbers (like 1/2, 4, and -3) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)). Together, these numbers encompass all possible values that can be found on a number line.
In our exercise, the notion of attempting to calculate the square root of \(-54\) highlights a key limitation within the real number system: it cannot accommodate the square roots of negative numbers. As mentioned earlier, no real number squared results in a negative value. Consequently, equations involving the square roots of negative numbers have no solutions within the real number system. Instead, these situations lead us towards complex numbers, which extend the real numbers to include solutions to these seemingly impossible problems.

Exploring Algebraic Solutions

Algebraic solutions involve finding the values of variables that satisfy given equations. When solving equations, such as the one in our exercise, you often manipulate the equation using algebraic operations to isolate the variable and find its value.
Our equation, \(x^2 + 7 = -47\), needed rearranging before attempting to solve for \(x\). Moving the constant \(7\) to the other side, we simplified it to \(x^2 = -54\). Traditionally, solving for \(x\) would involve taking the square root of both sides. But in this case, the equation results in taking the square root of a negative number, yielding no real algebraic solution. Instead, acknowledging that real solutions are not possible here is a crucial part of understanding algebraic problem-solving. This solution highlights that sometimes, an equation simply has no solution within a conventional number system.