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Variable substitution is a fundamental step when solving algebraic equations. It involves replacing a variable with a specific value to simplify the problem. This approach helps to focus on the remaining components of the equation.

For example, in the given equation \( x + 3 = x^2 - a \), we substitute \( x = -3 \). This means wherever we see \( x \), we replace it with \( -3 \). The purpose of this substitution is to transform the equation into a simpler form that is easier to manage.

After substitution, our equation becomes:

• Original: \( x + 3 = x^2 - a \)
• Substituted: \( -3 + 3 = (-3)^2 - a \)

This substituted equation simplifies our problem and leads us to the next steps of solving it.

Quadratic equations are polynomials of degree two, typically in the form \( ax^2 + bx + c = 0 \). These equations are fundamental in algebra and can have zero, one, or two solutions depending on the discriminant value.

In our specific case, the equation \( x + 3 = x^2 - a \) is manipulated to form a quadratic equation during variable substitution.

This equation is normally categorized among quadratic types due to the presence of the \( x^2 \) term. Recognizing it as a quadratic form can help determine the method of solving it, whether through factoring, completing the square, or using the quadratic formula.

However, after substitution with a specific \( x \) value, it no longer behaves as a quadratic, but understanding its structure is essential for handling similar problems.

Solving for variables is a critical skill in algebra that involves rearranging an equation to isolate the unknown variable. It requires a series of steps that simplify the equation.

In the problem, once the substitution \( x = -3 \) is applied, the equation simplifies to \( 0 = 9 - a \).

To solve for the variable 'a', we need to think of the equation as a balance scale. We perform operations equally on both sides to isolate 'a'.

Here, we can solve by:

• Identifying terms to move: We see \( a \) is on the right side of the equation. We want \( a \) alone.
• Rearranging: Subtract 9 from both sides so that \(- a = -9\).
• Isolating the variable: Multiply or divide accordingly to switch signs; here, this operation is straightforward, leading to \( a = 9 \).

This final answer gives us the value needed for the original condition to hold true.