Practice Writing And Solving Quadratic Equations.The Product Of Two Consecutive Integers Is 72. The Equation $x(x+1)=72$ Represents The Situation, Where $x$ Represents The Smaller Integer. Which Equation Can Be Factored And Solved For

Mar 1, 2025 by ADMIN 235 views

Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Understanding the Problem

In this article, we will practice writing and solving quadratic equations using a real-world problem. The problem states that the product of two consecutive integers is 72. We can represent this situation using the equation $x(x+1)=72$ , where $x$ represents the smaller integer. Our goal is to factor and solve this equation for $x$ .

Writing the Quadratic Equation

To write the quadratic equation, we need to expand the product of the two consecutive integers. We can do this by multiplying the two binomials:

x(x+1) = x^2 + x

Now, we can set this expression equal to 72:

x^2 + x = 72

This is a quadratic equation in the form $ax^2 + bx + c = 0$ , where $a=1$ , $b=1$ , and $c=-72$ .

Factoring the Quadratic Equation

To factor the quadratic equation, we need to find two numbers whose product is $ac$ and whose sum is $b$ . In this case, $ac = 1(-72) = -72$ , and $b = 1$ . We can find the two numbers by factoring -72 into two factors that add up to 1. However, since the product of the two numbers is negative, we need to consider the negative factors of 72.

The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

We can see that the factors 8 and -9 add up to -1, which is not correct. However, the factors -8 and 9 add up to 1, which is correct.

Therefore, we can factor the quadratic equation as:

(x+8)(x-9)=0

Solving the Quadratic Equation

To solve the quadratic equation, we need to set each factor equal to zero and solve for $x$ . We can do this by using the zero-product property, which states that if $ab=0$ , then either $a=0$ or $b=0$ .

Setting the first factor equal to zero, we get:

x+8=0

Subtracting 8 from both sides, we get:

x=-8

Setting the second factor equal to zero, we get:

x-9=0

Adding 9 to both sides, we get:

x=9

Therefore, the solutions to the quadratic equation are $x=-8$ and $x=9$ .

Conclusion

In this article, we practiced writing and solving quadratic equations using a real-world problem. We expanded the product of two consecutive integers, set it equal to 72, and factored the resulting quadratic equation. We then solved the quadratic equation by setting each factor equal to zero and solving for $x$ . The solutions to the quadratic equation are $x=-8$ and $x=9$ .

Practice Problems

Here are some practice problems to help you reinforce your understanding of quadratic equations:

The product of two consecutive integers is 120. Write and solve the quadratic equation to find the smaller integer.
A rectangle has a length of $x+5$ and a width of $x-3$ . The area of the rectangle is 48. Write and solve the quadratic equation to find the length of the rectangle.
A ball is thrown upward from the ground with an initial velocity of $x+2$ meters per second. The height of the ball is given by the equation $h(x) = -\frac{1}{2}x^2 + (x+2)$ . Write and solve the quadratic equation to find the time it takes for the ball to reach its maximum height.

Answer Key

Here are the answers to the practice problems:

The quadratic equation is $x(x+1)=120$ . The solutions are $x=-12$ and $x=10$ .
The quadratic equation is $(x+5)(x-3)=48$ . The solutions are $x=-8$ and $x=12$ .
The quadratic equation is $-\frac{1}{2}x^2 + (x+2) = 0$ . The solutions are $x=-4$ and $x=4$ .

Additional Resources

Here are some additional resources to help you learn more about quadratic equations:

Khan Academy: Quadratic Equations
Mathway: Quadratic Equations
Wolfram Alpha: Quadratic Equations

References

Here are some references to help you learn more about quadratic equations:

"Algebra and Trigonometry" by Michael Sullivan
"College Algebra" by James Stewart
"Quadratic Equations" by Michael Artin
Quadratic Equations Q&A ==========================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about quadratic equations. Whether you are a student, teacher, or just someone who wants to learn more about quadratic equations, this article is for you.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I write a quadratic equation?

A: To write a quadratic equation, you need to identify the variables and the constants in the problem. Then, you can use the general form of a quadratic equation to write the equation. For example, if the problem states that the product of two consecutive integers is 72, you can write the quadratic equation as $x(x+1)=72$ .

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is $ac$ and whose sum is $b$ . In this case, $ac = 1(-72) = -72$ , and $b = 1$ . You can find the two numbers by factoring -72 into two factors that add up to 1.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to set each factor equal to zero and solve for $x$ . You can do this by using the zero-product property, which states that if $ab=0$ , then either $a=0$ or $b=0$ .

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of $x$ that make the equation true. In other words, the solutions are the values of $x$ that satisfy the equation.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can only have two solutions. This is because the graph of a quadratic equation is a parabola, and a parabola can only intersect the x-axis at two points.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This is because the graph of a quadratic equation is a parabola, and a parabola can be above or below the x-axis. If the parabola is above the x-axis, then there are no solutions to the equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to use a graphing calculator or a computer program. You can also use a graphing app on your phone or tablet. To graph a quadratic equation, you need to enter the equation into the graphing tool and then adjust the window settings to see the graph.

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

A: Quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design bridges, buildings, and other structures.
Economics: Quadratic equations are used to model the behavior of economic systems.
Computer Science: Quadratic equations are used in computer graphics and game development.