In Exercises 115–122, find all values of x satisfying the given conditions. y1 = x - 1, y2 = x + 4 and y1y2 = 14

Verified step by step guidance

1. Identify the expressions for y1 and y2 given in the problem: y1 = x - 1 and y2 = x + 4.
2. Substitute the expressions for y1 and y2 into the equation y1y2 = 14: (x - 1)(x + 4) = 14.
3. Expand the left side of the equation using the distributive property: x^2 + 4x - x - 4 = 14.
4. Combine like terms and set the equation to zero: x^2 + 3x - 18 = 0.
5. Factor the quadratic equation or use the quadratic formula to find the values of x.

Key Concepts

Linear Equations

Linear equations are mathematical statements that express a relationship between two variables, typically in the form y = mx + b, where m is the slope and b is the y-intercept. In this question, y1 and y2 are linear equations representing two lines in a coordinate system. Understanding how to manipulate and solve these equations is essential for finding their intersection points.

Product of Functions

The product of functions involves multiplying two or more functions together to create a new function. In this case, y1y2 = 14 means that the product of the two linear functions must equal 14. This concept is crucial for setting up the equation that will allow us to find the values of x that satisfy both conditions simultaneously.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. When we set the product of the two linear functions equal to a constant (14), we can rearrange the equation into a quadratic form. Solving this quadratic equation will yield the values of x that satisfy the original conditions.