Not considering the sign of the quadratic expression

Oct 13, 2025 by windailys.com 27 views

Introduction

In mathematics, triangles are fundamental shapes used to model various real-world scenarios. Understanding the properties of triangles is crucial in solving problems related to geometry, physics, and engineering. In this article, we will explore the relationship between the base and height of a triangle and how to use inequalities to find the possible lengths of the base.

The Relationship Between Base and Height

Let's consider a triangle with a base of length $x$ inches and a height of $h$ inches. The problem states that the height of the triangle is 4 inches greater than twice its base. This can be expressed as:

h = 2x + 4

The Area of a Triangle

The area of a triangle is given by the formula:

A = \frac{1}{2}bh

where $b$ is the base and $h$ is the height. In this case, the area of the triangle is no more than 168 square inches, so we can write:

A \leq 168

Substituting the Relationship Between Base and Height

We can substitute the expression for $h$ in terms of $x$ into the area formula:

A = \frac{1}{2}bx

A = \frac{1}{2}b(2x + 4)

A = x^2 + 2x + 4

Setting Up the Inequality

Since the area of the triangle is no more than 168 square inches, we can set up the inequality:

x^2 + 2x + 4 \leq 168

Simplifying the Inequality

To simplify the inequality, we can subtract 168 from both sides:

x^2 + 2x - 164 \leq 0

Factoring the Quadratic Expression

The quadratic expression on the left-hand side can be factored:

(x + 13)(x - 12) \leq 0

Finding the Possible Values of $x$

To find the possible values of $x$ , we need to consider the sign of the quadratic expression. The expression is less than or equal to zero when one factor is positive and the other is negative. This occurs when $-13 \leq x \leq 12$ .

Conclusion

In conclusion, the inequality that can be used to find the possible lengths, $x$ , of the base of the triangle is:

(x + 13)(x - 12) \leq 0

This inequality represents the range of values for $x$ that satisfy the given conditions. By solving this inequality, we can determine the possible lengths of the base of the triangle.

Discussion

The problem presented in this article is a classic example of a quadratic inequality. The relationship between the base and height of the triangle is used to set up an inequality, which is then solved to find the possible values of the base. This type of problem is commonly encountered in mathematics and is used to model real-world scenarios.

Applications

The concept of quadratic inequalities has numerous applications in various fields, including:

Physics: Quadratic inequalities are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic inequalities are used to design and optimize systems, such as bridges and buildings.
Computer Science: Quadratic inequalities are used in algorithms and data structures to solve problems efficiently.

Final Thoughts

In this article, we explored the relationship between the base and height of a triangle and how to use inequalities to find the possible lengths of the base. The problem presented is a classic example of a quadratic inequality, and the solution involves factoring and solving the inequality. The concept of quadratic inequalities has numerous applications in various fields, and understanding this concept is crucial in solving problems related to geometry, physics, and engineering.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Q&A: The height of a triangle is 4 inches greater than twice its base. The area of the triangle is no more than 168 square inches. Which inequality can be used to find the possible lengths, $x$ , of the base of the triangle?

Q: What is the relationship between the base and height of the triangle?

A: The height of the triangle is 4 inches greater than twice its base. This can be expressed as:

h = 2x + 4

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

A: The area of a triangle is given by the formula:

A = \frac{1}{2}bh

where $b$ is the base and $h$ is the height.

Q: How do we set up the inequality to find the possible lengths of the base?

A: We can substitute the expression for $h$ in terms of $x$ into the area formula:

A = \frac{1}{2}bx

A = \frac{1}{2}b(2x + 4)

A = x^2 + 2x + 4

Since the area of the triangle is no more than 168 square inches, we can set up the inequality:

x^2 + 2x + 4 \leq 168

Q: How do we simplify the inequality?

A: To simplify the inequality, we can subtract 168 from both sides:

x^2 + 2x - 164 \leq 0

Q: How do we factor the quadratic expression?

A: The quadratic expression on the left-hand side can be factored:

(x + 13)(x - 12) \leq 0

Q: What are the possible values of $x$ ?

A: To find the possible values of $x$ , we need to consider the sign of the quadratic expression. The expression is less than or equal to zero when one factor is positive and the other is negative. This occurs when $-13 \leq x \leq 12$ .

Q: What is the final inequality that can be used to find the possible lengths, $x$ , of the base of the triangle?

A: The final inequality that can be used to find the possible lengths, $x$ , of the base of the triangle is:

(x + 13)(x - 12) \leq 0

Q: What are some real-world applications of quadratic inequalities?

A: Quadratic inequalities have numerous applications in various fields, including:

Physics: Quadratic inequalities are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic inequalities are used to design and optimize systems, such as bridges and buildings.
Computer Science: Quadratic inequalities are used in algorithms and data structures to solve problems efficiently.

Q: What are some resources for further learning on quadratic inequalities?

A: Some resources for further learning on quadratic inequalities include:

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra and Its Applications" by Gilbert Strang
[4] "Quadratic Equations and Inequalities" by Math Open Reference
[5] "Solving Quadratic Inequalities" by Khan Academy
[6] "Quadratic Inequalities in Real-World Applications" by Wolfram MathWorld

Q: What are some common mistakes to avoid when solving quadratic inequalities?

A: Some common mistakes to avoid when solving quadratic inequalities include:

Not considering the sign of the quadratic expression: Make sure to consider the sign of the quadratic expression when solving the inequality.
Not factoring the quadratic expression correctly: Make sure to factor the quadratic expression correctly to avoid errors.
Not considering the domain of the inequality: Make sure to consider the domain of the inequality to avoid errors.

Q: What are some tips for solving quadratic inequalities?

A: Some tips for solving quadratic inequalities include:

Use the quadratic formula to find the roots of the quadratic expression: The quadratic formula can be used to find the roots of the quadratic expression, which can help in solving the inequality.
Graph the quadratic expression to visualize the solution: Graphing the quadratic expression can help in visualizing the solution and understanding the behavior of the inequality.
Check the solution by plugging in values: Make sure to check the solution by plugging in values to ensure that it satisfies the inequality.