In Exercises 115–122, find all values of x satisfying the given conditions. y = 2x^2 - 3x and y = 2

Verified step by step guidance

1. Set the two equations equal to each other since they both equal y: 2x^2 - 3x = 2.
2. Subtract 2 from both sides to set the equation to zero: 2x^2 - 3x - 2 = 0.
3. Factor the quadratic equation. Look for two numbers that multiply to -4 (2 * -2) and add to -3.
4. Split the middle term using the numbers found in the previous step and factor by grouping.
5. Solve each factor set to zero to find the values of x.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. In this case, the function y = 2x^2 - 3x is a quadratic function where a = 2, b = -3, and c = 0. Understanding the properties of quadratic functions, such as their parabolas' shape and vertex, is essential for solving equations involving them.

Finding Intersections

To find the values of x that satisfy both equations, we need to determine the points of intersection between the two functions. This involves setting the two equations equal to each other (2x^2 - 3x = 2) and solving for x. The solutions represent the x-values where the graphs of the functions intersect, which is crucial for understanding their relationship.

Solving Quadratic Equations

Solving quadratic equations can be done using various methods, including factoring, completing the square, or applying the quadratic formula. In this context, after rearranging the equation to standard form, we may use the quadratic formula x = (-b ± √(b² - 4ac)) / (2a) to find the roots. Mastery of these techniques is vital for effectively solving the given problem.