The equation (x + 6)(x + 2) = 60 asks us to find the values of x that satisfy this equation. To find these values, we can use a variety of algebraic techniques, such as factoring, expanding, and isolating the variable x.
In this article, we will explore the different methods for solving the equation (x + 6)(x + 2) = 60. We will also discuss the applications of these methods in real-world scenarios.
Let’s get started by understanding the basic concepts of factoring and expanding algebraic expressions.
Solving Methods
Factoring the Expression
To factor the expression (x + 6)(x + 2), we can use the distributive property to expand it as follows:
(x + 6)(x + 2) = x^2 + 2x + 6x + 12 = x^2 + 8x + 12
We can then factor the resulting quadratic expression by finding two numbers that add up to 8 (the coefficient of x) and multiply to 12 (the constant term).
Using the Quadratic Formula
The quadratic formula can be used to solve a quadratic equation of the form ax^2 + bx + c = 0. In the case of the equation (x + 6)(x + 2) = 60, we have a = 1, b = 8, and c = 12.
Plugging these values into the quadratic formula, we get:
x = (-b ± sqrt(b^2 – 4ac)) / 2a
x = (-8 ± sqrt(8^2 – 4(1)(12))) / 2(1)
x = (-8 ± sqrt(16)) / 2
x = (-8 ± 4) / 2
x1 = -6 and x2 = -2
Completing the Square
Completing the square is another method for solving a quadratic equation. To complete the square, we take the following steps:
1. Move the constant term to the other side of the equation.
2. Divide both sides of the equation by the coefficient of x^2.
3. Add (b/2)^2 to both sides of the equation.
4. Factor the left-hand side of the equation as a perfect square.
5. Take the square root of both sides of the equation.
6. Solve for x.
Applications
Real-World Example
The equation (x + 6)(x + 2) = 60 has many applications in real-world scenarios. For example, it can be used to find the dimensions of a rectangle with a given area.
Suppose we want to find the dimensions of a rectangle with an area of 60 square units. Let x be the length of the rectangle and y be the width of the rectangle.
The area of a rectangle is given by the formula A = xy. Therefore, we can write the equation:
xy = 60
Because the rectangle has a length of x + 6 units and a width of x + 2 units, we can substitute these values into the equation as follows:
(x + 6)(x + 2) = 60
FAQ
What is the factored form of (x + 6)(x + 2)?
(x + 6)(x + 2) = x^2 + 8x + 12
What are the solutions to (x + 6)(x + 2) = 60?
x = -6 and x = -2
How can I use the quadratic formula to solve (x + 6)(x + 2) = 60?
x = (-8 ± sqrt(8^2 – 4(1)(12))) / 2(1)
What is completing the square?
Completing the square is a method for solving a quadratic equation by adding and subtracting a constant to the equation.
How can I use (x + 6)(x + 2) = 60 to find the dimensions of a rectangle?
Substitute the expressions for the length and width of the rectangle into the equation and solve for x.
Conclusion
In this article, we have explored the different methods for solving the equation (x + 6)(x + 2) = 60. We have also discussed the applications of these methods in real-world scenarios.
We hope this article has helped you to understand how to solve this type of equation. If you have any further questions, please feel free to leave a comment below.