Problem 5 - Algebra 2

Question

Simplify. $$ \left(x^{2}-4 x+3\right)-\left(4 x^{2}+3 x-5\right) $$

Step-by-Step Solution

Answer

-3x^2 - 7x + 8

Step 1: Distribute the Minus Sign

Start by distributing the minus sign to each term in the second bracket. This means changing the sign of every term inside the bracket:\[\left(x^{2} - 4x + 3\right) - \left(4x^{2} + 3x - 5\right) = x^{2} - 4x + 3 - 4x^{2} - 3x + 5\]

Step 2: Group Like Terms

Now group the like terms together:- Combine the \(x^2\) terms: \(x^2 - 4x^2\)- Combine the \(x\) terms: \(-4x - 3x\)- Combine the constant terms: \(3 + 5\)This yields:\[(x^2 - 4x^2) + (-4x - 3x) + (3 + 5)\]

Step 3: Combine Like Terms

Perform the operations for each group:- For the \(x^2\) terms: \(x^2 - 4x^2 = -3x^2\)- For the \(x\) terms: \(-4x - 3x = -7x\)- For the constants: \(3 + 5 = 8\)Putting it all together gives:\[-3x^2 - 7x + 8\]

Step 4: Write the Simplified Expression

The expression is now fully simplified as:\[-3x^2 - 7x + 8\]

Key Concepts

Distributive Property

When simplifying algebraic expressions, one of the most useful tools is the distributive property. This rule allows you to multiply a term with terms inside a parenthesis, distributing the multiplication across the terms within the brackets. It is often defined as: \( a(b + c) = ab + ac \). In our specific exercise, this property isn't used for multiplication directly but helps in distributing subtraction across terms. Because subtraction is essentially adding the opposite, when a negative sign is in front of a parenthesis, you distribute it to each term inside by changing their signs.
For example, given the expression \( -(4x^2 + 3x - 5) \), applying the distributive property changes it to \( -4x^2 - 3x + 5 \). This step is crucial as it ensures each term's sign is correctly accounted for when combined with terms from other brackets.

• Always remember to change each term's sign correctly.
• The distributive property equally applies to positive and negative multipliers.
• After distributing, each term stands on its own, ready for further simplification.

Understanding this property simplifies handling more complex algebraic expressions.

Combining Like Terms

Once the distributive property is applied, the next vital step is combining like terms. Like terms are terms that contain the same variables raised to the same power. The coefficients of these terms can be added or subtracted to simplify the expression.
In our exercise, after the distributive step, the expression becomes \( x^2 - 4x^2 - 4x - 3x + 3 + 5 \). Here, you group and simplify:

• Combine \( x^2 \) terms: \( x^2 - 4x^2 \) becomes \( -3x^2 \).
• Combine \( x \) terms: \( -4x - 3x \) becomes \( -7x \).
• Combine constant terms: \( 3 + 5 \) becomes \( 8 \).

The process of combining like terms is fundamental because:

• It reduces the number of terms, simplifying the expression.
• It ensures terms that can be combined are, preventing errors in further calculations.
• Each type of term (variables and constants) is dealt with separately, ensuring clarity.

Always check your work to ensure only like terms are combined.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and arithmetic operations. They are foundational in algebra, helping to model real-world situations and solve mathematical problems. Simplifying these expressions means reducing them to their simplest form by applying algebraic rules.
In our exercise, the expression \((x^{2}-4 x+3)-(4 x^{2}+3 x-5)\) is a clear example of an algebraic expression. Simplification involves:

• Applying distributive property and arithmetic operations.
• Combining like terms to make the expression more manageable.
• Resulting in simpler, fewer terms that can be computed quickly.

Simplifying is important for many reasons:

• It makes expressions easier to understand and work with.
• Helps solve equations by nurturing better algebra skills.
• Prepares for more complex operations in mathematics like solving and factoring equations.

Breaking down and simplifying algebraic expressions is a key skill, opening doors to solving almost any algebraic problem you may encounter.