Problem 51 - College Algebra with Corequisite Support

Question

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 \(\mathrm{ft} .^{2} .\) Solve the quadratic equation to find the length and width.

Step-by-Step Solution

Answer

The dimensions of the garden are 7 feet by 17 feet.

Step 1: Define variables

Let's denote the width of the garden as \( w \) (in feet). According to the problem, the length is 10 feet longer than the width. Therefore, the length is \( w + 10 \) feet.

Step 2: Express the area

The area of the garden is given as 119 square feet. The area of a rectangle is calculated by multiplying its length by its width. Thus, the equation for the area is: \( w(w + 10) = 119 \).

Step 3: Set up the quadratic equation

Simplify the area expression to form a quadratic equation:\[ w(w + 10) = 119 \]\[ w^2 + 10w = 119 \]

Step 4: Rearrange the equation

Rearrange the quadratic equation to standard form:\[ w^2 + 10w - 119 = 0 \]

Step 5: Solve the quadratic equation using the quadratic formula

The standard quadratic formula is \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for the equation \( ax^2 + bx + c = 0 \).Here, \( a = 1 \), \( b = 10 \), and \( c = -119 \).Calculate the discriminant: \( b^2 - 4ac = 10^2 - 4(1)(-119) = 100 + 476 = 576 \).Since the discriminant is positive, we have two real solutions:\[ w = \frac{-10 \pm \sqrt{576}}{2} \]\[ w = \frac{-10 \pm 24}{2} \]\( w = \frac{-10 + 24}{2} = 7 \) or \( w = \frac{-10 - 24}{2} = -17 \).

Step 6: Evaluate the solutions

We discard \( w = -17 \) because the width cannot be negative. Thus, the width \( w = 7 \) feet.

Step 7: Find the length

Using the width \( w = 7 \), find the length using \( w + 10 \).Length = 7 + 10 = 17 feet.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool used for solving quadratic equations. A quadratic equation is any equation that can be rewritten in the standard form \( ax^2 + bx + c = 0 \). In such equations, \( a \), \( b \), and \( c \) are constants and \( x \) represents an unknown variable that we need to solve for. The quadratic formula is given as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• The formula involves the symbol \( \pm \), which indicates that there can be two potential solutions.
• This formula is particularly useful because it can be applied to all types of quadratic equations, whether they have real or complex solutions.
• By plugging in the specific values of \( a \), \( b \), and \( c \) from the equation, we find the possible values for \( x \).

Remember, this formula helps us to bypass factoring, which sometimes can be complex or even impossible, depending on the equation. Therefore, it's an indispensable piece of algebra for anyone dealing with quadratic equations.

Area of a Rectangle

The area of a rectangle is a basic geometric concept that represents the amount of space within the boundary of a rectangle. It is calculated by multiplying the rectangle’s length by its width. In this situation for the garden, the length is given as 10 feet more than the width.

• Let’s denote the width as \( w \).
• The length is \( w + 10 \).
• Thus, the expression for the area is \( w(w + 10) \).
• Accordingly, the formula becomes \( Area = \, \, Length \times Width \).

In our example, this equation translates to \( w(w + 10) = 119 \). This leads directly to setting up a quadratic equation, necessary when you are solving for unknown dimensions of geometric figures with a given area.

Discriminant

The discriminant is a specific part of the quadratic formula, which helps predict the nature of the roots of the quadratic equation. The discriminant \( D \) is given by the expression under the square root:\[ b^2 - 4ac \]

• If \( D > 0 \), the quadratic equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution (or a repeated real root).
• If \( D < 0 \), there are no real solutions, but rather two complex solutions.

In this case, the discriminant was calculated as \( 576 \), indicating two real solutions. Understanding the discriminant's value before solving informs us about the nature of the solutions we expect, simplifying the deeper investigation into the roots.

Real Solutions

Real solutions refer to solutions that exist within the realm of real numbers, as opposed to complex numbers. When you solve a quadratic equation using the quadratic formula, the solutions can be real or complex based on the discriminant's value. Here, the solutions are found because our discriminant is positive.

• Two potential solutions appear due to the \( \pm \) sign involved in the quadratic formula.
• In our garden example, we calculated: \( w = \frac{-10 + 24}{2} = 7 \) or \( w = \frac{-10 - 24}{2} = -17 \).
• A meaningful check eliminates non-physical results like negative dimensions, discarding the solution \( w = -17 \).

This ensures the width of the garden is a feasible real number, resulting in \( w = 7 \) feet, making the length \( 17 \) feet. Real solutions ensure practical, usable figures in contexts such as geometry and other real-world applications.