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Verification by substitution is a method to check if a given value is a solution to an equation. This means substituting a proposed value into the equation to see if the equation holds true.

For example, let's verify if value ‘a = 5i’ is indeed a solution to the equation: \(x^2 + 25 = 0\).

First, substitute \(x\) with \(5i\) into the equation: \((5i)^2 + 25 = 0\).

Next, calculate \((5i)^2\); which is \(25i^2\), and since \(i^2 = -1\), we get \(25(-1)\).

Now, it becomes \(-25 + 25 = 0\), which simplifies to \(0 = 0\), a true statement!

Thus, \(x = 5i\) is indeed a solution.

Repeat the same process with \(x = -5i\). By following this method, you'll understand why the given value works as the solution.

The imaginary unit, represented by \(i\), is a fundamental concept in complex numbers. It is defined as \(i^2 = -1\).

This implies that \(i\) is the square root of -1. Imaginary numbers are numbers that involve this unit \(i\). When squared, \(i\) yields -1, allowing us to manage calculations involving negative square roots.

For instance, in the previous exercise, when squaring \(5i\), we get \((5i)^2\), which simplifies to \(25i^2\).

By knowing that \(i^2 = -1\), we convert it to \(25(-1)\), simplifying the problem.

Imaginary numbers are paired with real numbers to form complex numbers in the format \(a + bi\), where \(a\) and \(b\) are real numbers.

Solving quadratic equations involves finding values of \(x\) that satisfy the equation of the form \(ax^2 + bx + c = 0\).

When dealing with complex or imaginary numbers, the equation might need manipulation using the imaginary unit \(i\).

In the given exercise, we have the quadratic equation \(x^2 + 25 = 0\). This can be rewritten to find values of \(x\) such that \(x^2 = -25\).

By extracting the square roots of both sides, we get \(x = \pm \sqrt{-25}\).

Since \(\sqrt{-1} = i\), this reshapes to \(x = \pm 5i\).

Quadratic equations can often lead to complex solutions, especially when the determinant (\(b^2 - 4ac\)) is negative, involving \(i\). Techniques like factoring, using the quadratic formula, or completing the square are essential tools to find these solutions.