91Ó°ÊÓ

When solving equations, it's important to find what kind of solutions we are looking for. In this exercise, the focus is on 'real solutions'. Real solutions are values of the variable that make the equation true and are real numbers. Real numbers include all positive numbers, negative numbers, and zero. However, they do not include imaginary or complex numbers.

For example, if we consider the equation \[\frac{1}{x} + 2 = 3\], we are only looking for real values that make this equation true. The real value for this equation is when \(x = 1\). In summary, real solutions are the solutions that exist within the set of real numbers and no other type.

The square root function is a fundamental concept in mathematics. For any non-negative real number \(x\), the square root of \(x\) is a value \(y\) such that \(y^2 = x\). The symbol for the square root is \(\root\), and it is written as \(\root{x}\).

The key property of the square root function is that it always produces a non-negative output. This means that for any real number \(x\), \(\root{x} \geq 0\). For example, \(\root{16} = 4\) and \(\root{0} = 0\). Even if \(x\) is a perfect square, the square root of \(x\) will always be non-negative. This property is crucial for solving equations involving square roots, like the one in this exercise.

Understanding the non-negative output of the square root function is essential. This property tells us that the result of a square root function can never be negative. For all real numbers \(x\), the output of \(\root{x}\) will be zero or positive.

In the given equation \(\root{3t + 4} = -6\), we can immediately see a contradiction. Since the output of the square root function cannot be negative, it is impossible for \(\root{3t + 4}\) to equal \(-6\).

This is why the equation has no real solutions. No real number \(t\) can satisfy this equation because the fundamental properties of the square root function disallow negative outputs.

So, when encountering similar problems, remember that a square root result must always be zero or positive. This helps quickly identify whether an equation has a real solution or not.