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A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest power of the variable \( x \) is 2.
Understanding how to solve quadratic equations is essential in algebra because they frequently appear in various scientific and engineering problems. Common methods to solve a quadratic equation include:

• Factoring
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square

In the exercise, we're given a quadratic equation and tasked with substituting a potential solution to verify if it satisfies the equation. This verification involves multiple algebraic manipulations, demonstrating one way to approach solutions beyond traditional solving methods.

Radical expressions involve roots, such as square roots, cube roots, and so on. In our exercise, \( \sqrt{20} \) is a radical expression.
Handling radical expressions requires special attention to ensure all operations abide by the laws of exponents and roots:

• Simplifying radicals involves finding the largest square factor. For example, \( \sqrt{20} \) can be simplified to \( 2\sqrt{5} \) because 20 = 4 × 5, and \( \sqrt{4} = 2 \).
• When dealing with sums or differences involving radicals, such as \( (6 - \sqrt{20}) \), careful expansion and simplification are crucial when squaring these expressions.
• Always perform operations step by step to maintain accuracy, especially during substitution as seen in the original solution.

In our exercise, correctly expanding and simplifying radical expressions is necessary to show that the proposed solution satisfies the equation.

The substitution method involves replacing a variable in an equation with a given expression, then simplifying to see if the equation holds true. This method is particularly useful when verifying solutions.
When dealing with quadratic equations, substitution is used as follows:

• Identify a potential solution, in this case, \( x = 6 - \sqrt{20} \).
• Replace every instance of \( x \) in the equation with the expression for \( x \) given as the potential solution.
• Follow through with the algebraic manipulations: expand, simplify, and combine like terms to evaluate whether the equation resolves correctly.

The substitution method is powerful for checking solutions because it straightforwardly incorporates into many types of algebraic equations, allowing for confirmation of complex expressions such as those in the given quadratic scenario.