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Chapter 5: Problem 93

Perform the operations and simplify. $$4 x-3(x-5)^{2}$$

Short Answer

Expert verified
-3x^2 + 34x - 75

Step by step solution

01

Expand the expression

Start by expanding the term \( -3(x-5)^2 \). Use the distributive property and the formula for squaring a binomial: \( (a - b)^2 = a^2 - 2ab + b^2 \), where \( a = x \) and \( b = 5 \).
02

Calculate the binomial square

Square the binomial \( x-5 \) to get \[ (x-5)^2 = x^2 - 10x + 25 \].
03

Distribute the -3

Distribute \( -3 \) through the expanded binomial expression: \[ -3(x^2 - 10x + 25) = -3x^2 + 30x - 75 \].
04

Combine like terms

Combine the expression \( 4x \) with the expanded and distributed term: \[ 4x + (-3x^2 + 30x - 75) \]. This simplifies to \[ -3x^2 + 34x - 75 \].
05

Simplified expression

The simplified expression is \[ -3x^2 + 34x - 75 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra. It allows us to multiply a single term across terms inside parentheses. For example, in the expression \(-3(x-5)^2\), the \(-3\) will be distributed to each term inside the expanded parentheses. Here鈥檚 how it works: \[a(b + c) = ab + ac\]. This property simplifies complex expressions and is key to solving algebra problems efficiently.
Binomial Expansion
Understanding binomial expansion, especially when squaring binomials, is crucial. When you expand \((x - 5)^2\), you're essentially using the formula \((a - b)^2 = a^2 - 2ab + b^2\) where \(a = x\) and \(b = 5\). Following this, \((x - 5)^2 = x^2 - 10x + 25\). Each term within the binomial is squared, first multiplied by itself, then by each other, and lastly summed.
Like Terms
Combining like terms is an essential step to simplify algebraic expressions. Like terms have the same variable raised to the same power. In the final step of our problem, \(4x\) and \(30x\) are like terms because both are linear terms with the variable \(x\). They combine to form \(34x\). This consolidation of like terms leads to a simplified expression: \(-3x^2 + 34x - 75\). Such simplifications make it easier to solve and understand the expression.

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Most popular questions from this chapter

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