Problem 27 - Algebra A Combined Function

Question

If the sides of a square are increased by 3 inches, the area becomes 64 square inches. Find the length of the sides of the original square.

Step-by-Step Solution

Answer

The original side length is 5 inches.

Step 1: Define the Problem

Let's denote the length of the side of the original square as \( x \). If the sides increase by 3 inches, the new side length becomes \( x + 3 \). The problem states that this new square has an area of 64 square inches.

Step 2: Set Up the Equation for Area

The area of a square is given by the formula \( \text{side}^2 \). For the new square, this becomes \((x + 3)^2 = 64\). We need to solve this equation to find \( x \), the side length of the original square.

Step 3: Solve the Equation

Expand the equation: \((x + 3)^2 = x^2 + 6x + 9\). So, the equation \( x^2 + 6x + 9 = 64 \) needs to be solved. First, subtract 64 from both sides to set the equation to zero: \( x^2 + 6x + 9 - 64 = 0 \), which simplifies to \( x^2 + 6x - 55 = 0 \).

Step 4: Factor the Quadratic Equation

Now, let's factor the quadratic equation \( x^2 + 6x - 55 = 0 \). We are looking for two numbers whose product is \(-55\) and sum is \(6\). These numbers are \(11\) and \(-5\), so the equation can be factored as \((x + 11)(x - 5) = 0\).

Step 5: Solve for \( x \)

Set each factor equal to zero: \(x + 11 = 0\) and \(x - 5 = 0\). Solving these gives \(x = -11\) and \(x = 5\). Since a square's side length can't be negative, \(x = -11\) is not valid. Therefore, the length of the side of the original square is \(x = 5\) inches.

Key Concepts

Understanding Quadratic Equations

Quadratic equations might sound complex, but they are just a type of polynomial equation. A standard quadratic equation takes the form of \( ax^2 + bx + c = 0 \). The highest exponent here is 2, which is why we call it "quadratic." Here, each letter represents a coefficient or constant. For example, in our exercise, the equation \( x^2 + 6x - 55 = 0 \) is a quadratic equation. This represents the problem where an unknown quantity (\( x \)) needs to be solved.

To solve quadratic equations, you have several methods available:

• Factoring - Breaking down the equation into simpler terms that multiply to make the original equation.
• Using the Quadratic Formula - \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides the solutions directly.
• Completing the Square - Adding terms to both sides of the equation to make a perfect square trinomial.

Different equations will require different methods, depending on what's easiest or quickest in the situation.

The Art of Factoring

Factoring is a key technique in algebra used to simplify equations. When you factor, you're breaking down an expression into a product of simpler expressions. In the context of quadratic equations, factoring helps us find the roots, or solutions, that make the equation true. Let’s see how you can apply this.

In our exercise, the equation \( x^2 + 6x - 55 = 0 \) was factored into \((x + 11)(x - 5) = 0\). The idea is to express the quadratic in terms of two binomials like we did here.

• Search for two numbers: These two numbers need to multiply to \(-55\) (the constant term) and add up to \(6\) (the coefficient of \(x\)).
• Break it down: Once you know these numbers (\(11\) and \(-5\)), they help you write the equation as a product of binomial factors.
• Find solutions: Set each factor equal to zero, solving for \(x\). This gives the possible values for \(x\).

Factoring is handy and often provides a speedy way to solve quadratic equations, as seen in this exercise.

Calculating the Area of a Square

In geometry, a square is a special case of a rectangle where all sides are equal. This makes calculating the area very straightforward. The area \(A\) of a square with a side length \(s\) is given by the formula \( A = s^2 \).

For instance, in our exercise, the area of the new square was described by \((x + 3)^2 = 64\). Here's how this relates:

• The expression \((x + 3)^2\) demonstrates the algebraic representation of a square’s area based on its side length.
• The equation \((x + 3)^2 = 64\) sets up a situation where a known area helps us backtrack to find the side length.
• This means solving \(x + 3 = 8\) after taking the square root of 64, which helps us find the new side length of the square, allowing us to then determine the original side length, \(x\).

Understanding the relationship between side length and area in a square helps you tackle similar problems effortlessly.