Problem 27 - Algebra A Combined Function

Question

Subtract. $$ \left(0.7 x^{2}+0.2 x-0.8\right)-\left(0.9 x^{2}+1.4\right) $$

Step-by-Step Solution

Answer

\(-0.2x^2 + 0.2x - 2.2\)

Step 1: Distribute the Negative Sign

The problem involves subtracting two expressions: \( (0.7x^2 + 0.2x - 0.8) - (0.9x^2 + 1.4) \). First, distribute the negative sign to each term in the second polynomial. This gives us \( 0.7x^2 + 0.2x - 0.8 - 0.9x^2 - 1.4 \).

Step 2: Combine Like Terms

Next, identify and combine the like terms:- Combine the \(x^2\) terms: \(0.7x^2 - 0.9x^2 = -0.2x^2\).- Combine the constant terms: \(-0.8 - 1.4 = -2.2\).- The \(x\) term \(0.2x\) has no like term, so it stays as it is.

Step 3: Write the Simplified Expression

After combining the like terms, the resulting expression is \(-0.2x^2 + 0.2x - 2.2\).

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra that helps simplify expressions by allowing you to multiply a term over a set of terms within parentheses. In the context of polynomial subtraction, distributing often means distributing a negative sign across terms in the polynomial you are subtracting. For example, consider the expression \((A - B)\). To subtract \(B\) from \(A\), you transform this into \(A + (-1) \cdot B\). This step is vital because it ensures that each term in \(B\) is subtracted correctly.In our specific exercise, we began with: \((0.7x^2 + 0.2x - 0.8) - (0.9x^2 + 1.4)\). We applied the distributive property by multiplying each term in the second expression by \(-1\), which turned it into:

• \(+0.7x^2 + 0.2x - 0.8\)
• \(-0.9x^2\)
• \(-1.4\)

This step effectively turned the expression into an addition problem, making subsequent steps much simpler.

Combining Like Terms

Combining like terms is another crucial step in simplifying expressions. It involves identifying terms in a polynomial that have similar variables raised to the same power and adding or subtracting their coefficients. This process helps to condense the expression, making it easier to understand and solve.For instance, in our problem, the expression after distribution was:\(0.7x^2 + 0.2x - 0.8 - 0.9x^2 - 1.4\).To combine like terms:

• Look at the \(x^2\) terms: \(0.7x^2\) and \(-0.9x^2\). By subtracting their coefficients, we get \(-0.2x^2\).
• The term \(0.2x\) has no other \(x\) terms to combine with, so it remains \(0.2x\).
• Combine the constants \(-0.8\) and \(-1.4\), which add up to \(-2.2\).

By organizing similar terms together, you reduce clutter within the equation and reveal the most simplified form of the polynomial.

Simplifying Expressions

Simplifying expressions combines different methods, such as distributing and combining like terms, to arrive at the most concise version of an algebraic expression.After applying distribution and combining like terms, the expression \[(0.7x^2 + 0.2x - 0.8) - (0.9x^2 + 1.4)\] became \(-0.2x^2 + 0.2x - 2.2\). This final expression is much simpler:

• The polynomial is reduced to only necessary terms.
• Each component is clearly represented: one quadratic term \(-0.2x^2\), one linear term \(0.2x\), and one constant \(-2.2\).

An expression in its simplified form is easier to handle once you advance to solving equations or further algebraic manipulations. Whether you're graphing, factoring, or setting up equations, simplified forms help you tackle each problem with clarity and confidence.