Problem 28 - Algebra A Combined Function

Question

Subtract. $$ \left(-0.3 y^{2}+0.6 y-0.3\right)-\left(0.5 y^{2}+0.3\right) $$

Step-by-Step Solution

Answer

The result of the subtraction is \(-0.8 y^{2} + 0.6 y - 0.6\).

Step 1: Distribute the Negative Sign

In the expression \(\left(-0.3 y^{2}+0.6 y-0.3\right)-\left(0.5 y^{2}+0.3\right)\), we first distribute the negative sign across the second group. This changes the expression to:\[-0.3 y^{2}+0.6 y-0.3 - 0.5 y^{2} - 0.3\]

Step 2: Combine Like Terms

Now combine the like terms:- The \(y^2\) terms: \(-0.3 y^{2} - 0.5 y^{2} = -0.8 y^{2}\)- The constant terms: \(-0.3 - 0.3 = -0.6\)The \(y\) term remains as is since there are no like terms to combine. The expression becomes:\[-0.8 y^{2} + 0.6 y - 0.6\]

Step 3: Final Simplified Expression

The simplified expression after the subtraction is complete is:\[-0.8 y^{2} + 0.6 y - 0.6\]

Key Concepts

Understanding Like Terms

In algebra, combining like terms is an essential skill for simplifying expressions and solving equations. Like terms are terms within an algebraic expression that have identical variable components, including their exponents. This means that for terms to be 'like', they must have exactly the same variable and power, but can have different numerical coefficients. For example, in the expression

• \(-0.3y^2\)
• \(0.5y^2\)

both terms are 'like' terms because they each contain \(y^2\) as the variable component. Thus, they can be combined by adding or subtracting their coefficients. In our exercise, to combine these terms, you subtract the coefficients:

• \(-0.3 - 0.5 = -0.8\)

so you end up with \(-0.8y^2\). Remember that terms such as \(0.6y\) and constants like \(-0.3\) are not like terms since they do not share the same variable structure. This is why they are left unchanged when combining other terms.

Distribute the Negative Sign

The operation of subtraction in algebra is often approached by "distributing the negative sign." This method helps prevent common errors that occur when dealing with negative numbers. Distributing a negative sign across an entire expression means changing all of the signs in that expression. Looking back at our original problem, we started with:

• \((-0.3y^2 + 0.6y - 0.3) - (0.5y^2 + 0.3)\)

To subtract the second expression, you must first distribute the negative sign across the entire expression in parentheses. This step transforms each term in the parentheses from positive to negative, or vice versa:

• \(0.5y^2\) becomes \(-0.5y^2\)
• \(0.3\) becomes \(-0.3\)

Converting these terms gives us:

• \(-0.3y^2 + 0.6y - 0.3 - 0.5y^2 - 0.3\)

This process ensures that the subtraction is carried out properly regardless of the complexity of the expression.

Simplified Expression

Reaching a simplified expression means the expression is reduced to its simplest form, with no like terms left to combine and no unnecessary parentheses. In algebra, simplification results in an expression that is much easier to understand and use in further calculations. Let’s apply this to our exercise:

• Combine like terms: \(-0.3y^2 - 0.5y^2 = -0.8y^2\)
• The \(0.6y\) component remains unchanged because it has no like terms in the expression.
• Combine constants: \(-0.3 - 0.3 = -0.6\)

Thus, the fully simplified expression becomes:

• \(-0.8y^2 + 0.6y - 0.6\)

This form is not only concise, but also ready for any further algebraic operations that may be required in problem-solving. Always remember, a truly simple expression is a powerful tool in algebra.