Solve The Equation:${ X(x + 2) = 72 } S I M P L I F Y A N D R E A R R A N G E T O F O R M A Q U A D R A T I C E Q U A T I O N : Simplify And Rearrange To Form A Quadratic Equation: S Im Pl I F Y An D Re A Rr An G E T O F Or Ma Q U A D R A T I Ce Q U A T I O N : { X^2 + 2x - 72 = 0 \}

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will delve into the world of quadratic equations and provide a step-by-step guide on how to solve them. We will start with a simple equation and work our way up to more complex ones.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x) 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.

Solving the Equation

Let's start with the equation we were given:

x(x + 2) = 72

To simplify this equation, we can expand the left-hand side using the distributive property:

x^2 + 2x = 72

Now, we can rearrange the equation to form a quadratic equation:

x^2 + 2x - 72 = 0

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

Factoring the Quadratic Equation

One way to solve a quadratic equation is to factor it. Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can factor the quadratic equation as follows:

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

This tells us that either (x + 9) = 0 or (x - 8) = 0.

Solving for x

Now that we have factored the quadratic equation, we can solve for x by setting each factor equal to zero:

x + 9 = 0 --> x = -9

x - 8 = 0 --> x = 8

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

Using the Quadratic Formula

Another way to solve a quadratic equation is to use the quadratic formula. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

In this case, we have a = 1, b = 2, and c = -72. Plugging these values into the quadratic formula, we get:

x = (-(2) ± √((2)^2 - 4(1)(-72))) / 2(1)

x = (-2 ± √(4 + 288)) / 2

x = (-2 ± √292) / 2

x = (-2 ± √(4(73))) / 2

x = (-2 ± 2√73) / 2

x = -1 ± √73

Therefore, the solutions to the quadratic equation are x = -1 + √73 and x = -1 - √73.

Conclusion

Solving quadratic equations is a crucial skill in mathematics, and there are several ways to do it. In this article, we have discussed two methods: factoring and using the quadratic formula. We have also provided a step-by-step guide on how to solve a quadratic equation using both methods.

Tips and Tricks

• When factoring a quadratic equation, look for two numbers whose product is equal to the constant term (in this case, -72) and whose sum is equal to the coefficient of the linear term (in this case, 2).
• When using the quadratic formula, make sure to plug in the correct values for a, b, and c.
• When solving for x, make sure to simplify the expression and check for any extraneous solutions.

Real-World Applications

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 and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Final Thoughts

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x) is two. The general form of a quadratic equation is:

ax^2 + bx + c = 0

Q: How do I solve a quadratic equation?

A: There are several ways to solve a quadratic equation, including factoring, using the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression and solve for x.

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

A: Factoring involves expressing the quadratic equation as a product of two binomials, while using the quadratic formula involves plugging in the values of a, b, and c into the formula.

Q: When should I use factoring and when should I use the quadratic formula?

A: You should use factoring when the quadratic equation can be easily factored, and you should use the quadratic formula when the quadratic equation cannot be easily factored.

Q: How do I check my solutions?

A: To check your solutions, plug the values of x back into the original equation and simplify. If the equation is true, then the value of x is a valid solution.

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

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

• Not simplifying the expression
• Not checking for extraneous solutions
• Not using the correct values for a, b, and c
• Not following the correct order of operations

Q: How do I apply quadratic equations to real-world problems?

A: Quadratic equations have many real-world applications, including physics, engineering, and economics. To apply quadratic equations to real-world problems, you need to identify the variables and constants in the problem, and then use the quadratic formula or factoring to solve for the unknown variable.

Q: What are some examples of quadratic equations in real-world problems?

A: Some examples of quadratic equations in real-world problems include:

• Modeling the motion of objects under the influence of gravity
• Designing and optimizing systems, such as bridges and buildings
• Modeling the behavior of economic systems, such as supply and demand

Q: How do I practice solving quadratic equations?

A: To practice solving quadratic equations, you can try solving problems on your own, or use online resources, such as worksheets and practice tests. You can also try solving quadratic equations with different coefficients and constants to see how the solutions change.

Q: What are some advanced topics in quadratic equations?

A: Some advanced topics in quadratic equations include:

• Complex numbers
• Quadratic equations with complex coefficients
• Quadratic equations with non-integer coefficients
• Quadratic equations with non-linear terms

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have many real-world applications. By following the step-by-step guide provided in this article, you can learn how to solve quadratic equations using both factoring and the quadratic formula. Remember to always simplify the expression and check for any extraneous solutions. With practice and patience, you can become proficient in solving quadratic equations and apply them to real-world problems.