The Area Of A Rectangular Garden Is $84 \, \text{ft}^2$. The Length Of The Garden Is 5 Ft More Than The Width Of The Garden. This Situation Can Be Represented By The Equation $w^2 + 5w - 84 = 0$. What Is The Width Of The Garden?A.

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Introduction

In this article, we will explore the problem of finding the width of a rectangular garden given its area and the relationship between its length and width. The area of the garden is given as $84 , \text{ft}^2$, and the length is 5 ft more than the width. This situation can be represented by a quadratic equation, which we will solve to find the width of the garden.

The Quadratic Equation

The quadratic equation that represents the situation is $w^2 + 5w - 84 = 0$, where $w$ is the width of the garden. This equation can be factored or solved using the quadratic formula.

Factoring the Quadratic Equation

To factor the quadratic equation, we need to find two numbers whose product is $-84$ and whose sum is $5$. These numbers are $12$ and $-7$, since $12 \times (-7) = -84$ and $12 + (-7) = 5$. Therefore, we can write the quadratic equation as:

w2+5w−84=(w+12)(w−7)=0w^2 + 5w - 84 = (w + 12)(w - 7) = 0

Solving the Quadratic Equation

Now that we have factored the quadratic equation, we can set each factor equal to zero and solve for $w$.

w+12=0⇒w=−12w + 12 = 0 \Rightarrow w = -12

w−7=0⇒w=7w - 7 = 0 \Rightarrow w = 7

The Width of the Garden

Since the width of the garden cannot be negative, we can discard the solution $w = -12$. Therefore, the width of the garden is $w = 7$ ft.

Conclusion

In this article, we have solved a quadratic equation to find the width of a rectangular garden given its area and the relationship between its length and width. We have factored the quadratic equation and solved for the width, which is $w = 7$ ft.

Additional Information

Quadratic Equations

Quadratic equations are a type of polynomial equation of degree two, which means the highest power of the variable is two. They have the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using factoring, the quadratic formula, or other methods.

Factoring Quadratic Equations

Factoring quadratic equations involves expressing the equation as a product of two binomials. This can be done by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

The Quadratic Formula

The quadratic formula is a method for solving quadratic equations that involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

References

Keywords

  • Quadratic equation
  • Factoring
  • Quadratic formula
  • Rectangular garden
  • Area
  • Length
  • Width

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Introduction

In our previous article, we solved a quadratic equation to find the width of a rectangular garden given its area and the relationship between its length and width. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is $A = lw$, where $A$ is the area, $l$ is the length, and $w$ is the width.

Q: How do I find the length of the garden if I know the width?

A: If you know the width of the garden, you can find the length by using the formula $l = \frac{A}{w}$, where $A$ is the area and $w$ is the width.

Q: Can I use the quadratic formula to solve this problem?

A: Yes, you can use the quadratic formula to solve this problem. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the difference between factoring and the quadratic formula?

A: Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions to the equation.

Q: Can I use the quadratic formula if the equation does not factor easily?

A: Yes, you can use the quadratic formula even if the equation does not factor easily. The quadratic formula is a general method for solving quadratic equations.

Q: What if I have a quadratic equation with complex solutions?

A: If you have a quadratic equation with complex solutions, you can use the quadratic formula to find the solutions. The quadratic formula will give you two complex solutions.

Q: Can I use the quadratic formula to solve quadratic equations with fractional coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with fractional coefficients. The quadratic formula will give you the solutions to the equation.

Additional Information

Quadratic Equations with Complex Solutions

Quadratic equations with complex solutions can be solved using the quadratic formula. The quadratic formula will give you two complex solutions.

Quadratic Equations with Fractional Coefficients

Quadratic equations with fractional coefficients can be solved using the quadratic formula. The quadratic formula will give you the solutions to the equation.

Factoring Quadratic Equations with Complex Solutions

Factoring quadratic equations with complex solutions can be challenging. In some cases, it may be easier to use the quadratic formula to find the solutions.

References

Keywords

  • Quadratic equation
  • Factoring
  • Quadratic formula
  • Rectangular garden
  • Area
  • Length
  • Width
  • Complex solutions
  • Fractional coefficients