Problem 53 - 1,001 GED Practice Questions For Dummies

Question

\- \((x+y)^2\) is equivalent to which of the following? (A) \(x^2+y^2\) (B) \(x^2 y^2\) (C) \(x^2+y^2+2 x y\) (D) \(2 x+2 y\)

Step-by-Step Solution

Answer

The correct answer is (C) \(x^2+y^2+2 x y\).

Step 1: Expanding the expression \((x+y)^2\) using the distributive property (FOIL method)

The FOIL method stands for First, Outer, Inner, and Last, representing the different pairs of terms multiplied together. We start by writing down the given expression and applying FOIL: \[(x+y)^2 = (x+y)(x+y)\] First terms: \(x \cdot x = x^2\) Outer terms: \(x \cdot y = x y\) Inner terms: \(y \cdot x = y x\) Last terms: \(y \cdot y = y^2\)

Step 2: Combine like terms

Now, we combine the like terms: \(x^2 + xy + yx + y^2 = x^2 + x y + y x + y^2 = x^2 + 2 x y + y^2\)

Step 3: Compare the expanded expression to the given options

We have expanded the expression and found the equivalent expression is \(x^2 + 2 x y + y^2\). Now, let's compare our result to the available options: (A) \(x^2+y^2\) - This option doesn't include the term \(2 x y\). (B) \(x^2 y^2\) - This option is an incorrect expansion. (C) \(x^2+y^2+2 x y\) - This option is the same as our expanded expression. (D) \(2 x+2 y\) - This option is an incorrect expansion.

Step 4: Choose the correct answer

Based on our comparison, we can now conclude that the correct answer is: \((x+y)^2\) is equivalent to (C) \(x^2+y^2+2 x y\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a staple of algebra and serve as an important foundation for higher-level maths. A quadratic equation is typically presented in the form of \( ax^2 + bx + c = 0 \) where \( a \), \( b \) and \( c \) are known values, and \( a \) is not zero. The graph of a quadratic equation is a parabola that either opens upwards or downwards, depending on the sign of \( a \).

Quadratic equations are notable for their appearance in various applications, including physics, engineering, and economics. To solve these equations, one could use factoring, completing the square, using the quadratic formula, or graphing. When practicing for the GED math tests, understanding how to manipulate and solve quadratic equations is crucial as it's a recurring concept.

Let's examine why the quadratic nature of \( (x+y)^2 \) is more than a simple expression. By expanding this expression, we can reveal its quadratic nature, which includes the squared terms \( x^2 \) and \( y^2 \) and the interaction term \( 2xy \) representing the product of the variables. Recognizing these components is essential to mastering quadratic equations.

Demystifying the FOIL Method

The FOIL method is a process used to multiply two binomials and is essential to handling algebraic expressions like quadratic equations. FOIL stands for First, Outer, Inner, and Last, indicating the order in which you multiply the terms in the binomials.

Let's apply it: For \( (x+y)^2 = (x+y)(x+y) \), the First terms are the first terms of each binomial, so \( x \cdot x = x^2 \) in our case. For the Outer terms, we multiply the outermost terms, hence \( x \cdot y = xy \). The Inner terms are \( y \cdot x = yx \) (notice that \( xy \) is the same as \( yx \) due to the commutative property of multiplication). Lastly, the Last terms are the last terms of each binomial, yielding \( y \cdot y = y^2 \).

By combining these, you get \( x^2 + xy + yx + y^2 \) which simplifies to \( x^2 + 2xy + y^2 \). The FOIL method is a reliable tool that, when practiced, becomes second nature, facilitating the simplification of many algebraic expressions on your GED math practice tests.

Applying the Distributive Property

The distributive property is fundamental in algebra and offers a systematic way to expand expressions, such as the product of two binomials. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. This property is written as \( a(b+c) = ab + ac \).

When we deal with an expression like \( (x+y)^2 \), we are using the distributive property to expand it. Here's how it works: \( (x+y)(x+y) \) can be seen as \( x \) times the sum \( (x+y) \), plus \( y \) times the sum \( (x+y) \). Thus, you distribute the \( x \) and \( y \) across the \( x+y \) to get \( x^2 + xy + yx + y^2 \) or, simplified to \( x^2 + 2xy + y^2 \).

The distributive property is omnipresent in math and is employed in simplifying expressions, combining like terms, and solving equations. When practicing GED math, students should focus on mastering this property as it is a gateway to tackling complex problems with confidence.