Problem 61 - Elementary Algebra

Question

State the restrictions and then simplify. $$ x 2-4(2-x) 2 $$

Step-by-Step Solution

Answer

Restrictions: None. Simplified expression: \(-3x^2 + 16x - 16\).

Step 1: Identify Parts of the Expression

The expression given is \(x^2 - 4(2-x)^2\). Notice that it consists of a quadratic term \(x^2\) and a term that needs expansion, \(-4(2-x)^2\).

Step 2: Expand the Squared Term

Expand the squared term \((2-x)^2\). Using the identity \((a-b)^2 = a^2 - 2ab + b^2\), it becomes: \((2)^2 - 2(2)(x) + (x)^2 = 4 - 4x + x^2\).

Step 3: Multiply the Expansion by -4

Now, multiply the expanded term \(4 - 4x + x^2\) by -4: \(-4(4 - 4x + x^2) = -16 + 16x - 4x^2\).

Step 4: Combine like terms

Now, rewrite the entire expression with the expanded and multiplied term: \(x^2 - 16 + 16x - 4x^2\). Combine like terms \(x^2 - 4x^2 = -3x^2\) and remain with \(-3x^2 + 16x - 16\).

Step 5: State the Restrictions

In this problem, we assume no restrictions on \(x\) because no division or square roots are introduced. Therefore, \(x\) is any real number.

Key Concepts

Expansion of Binomials

The expansion of binomials is a key skill in algebra that involves expanding an expression of the form \( (a + b)^n \).For our exercise, we focus on expanding a squared binomial \( (2-x)^2 \).The formula used is:

• \( (a-b)^2 = a^2 - 2ab + b^2 \).

This pattern helps us break down the expression step-by-step:

• First, calculate \( a^2 \), which is \( 2^2 = 4 \).
• Next, calculate \( -2ab \), which is \( -2 \times 2 \times x = -4x \).
• Finally, calculate \( b^2 \), which is \( x^2 \).

Combining these results gives us the expanded form: \( 4 - 4x + x^2 \). Expansion allows us to simplify complex expressions and make further operations like multiplication or addition easier.

Combining Like Terms

Combining like terms is all about putting like terms together. In algebra, like terms are terms that have the same variable raised to the same power.For this problem, we processed the expression after expansion and multiplication as:

• \( x^2 - 16 + 16x - 4x^2 \).

We identify and group terms of the same type:

• \( x^2 \) and \( -4x^2 \) are like terms.
• Combine them to get \( -3x^2 \) (i.e., \( x^2 - 4x^2 \)).

Putting it all together, the simplified expression becomes \( -3x^2 + 16x - 16 \).Combining like terms simplifies the expression, making it more concise and easier to work with for solving or further calculations.

Quadratic Expressions

Quadratic expressions play a significant role in algebra, typically appearing in the form \( ax^2 + bx + c \).Our final expression, \( -3x^2 + 16x - 16 \), is a quadratic expression:

• The term \(-3x^2\) is the quadratic term, with the coefficient \(a = -3\).
• The linear term \(16x\) has coefficient \(b = 16\).
• The constant term is \(-16\), with \(c = -16\).

Quadratic expressions can be represented graphically as parabolas.They are essential for solving real-world problems involving area, projectile motion, and more. Understanding their structure helps in predicting and analyzing their behavior effectively.From this exercise, we crafted a quadratic expression by expanding, multiplying, and combining terms.