In Exercises 115–122, find all values of x satisfying the given conditions. y1 = 2x^2 + 5x - 4, y2 = - x^2 + 15x - 10, and y1 - y2 = 0

Verified step by step guidance

1. Start by setting up the equation y1 - y2 = 0, which means 2x^2 + 5x - 4 - (-x^2 + 15x - 10) = 0.
2. Simplify the equation by distributing the negative sign: 2x^2 + 5x - 4 + x^2 - 15x + 10 = 0.
3. Combine like terms: (2x^2 + x^2) + (5x - 15x) + (-4 + 10) = 0.
4. This simplifies to 3x^2 - 10x + 6 = 0.
5. Solve the quadratic equation 3x^2 - 10x + 6 = 0 using the quadratic formula, factoring, or completing the square.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax^2 + bx + c. They graph as parabolas and can have various properties such as vertex, axis of symmetry, and roots. Understanding how to manipulate and analyze these functions is crucial for solving equations involving them.

Finding Intersections

Finding the intersection points of two functions involves setting them equal to each other and solving for the variable. In this case, we set y1 equal to y2 to find the values of x where the two parabolas intersect. This process often leads to solving a quadratic equation, which can yield multiple solutions.

Solving Quadratic Equations

Solving quadratic equations can be done using various methods such as factoring, completing the square, or applying the quadratic formula. Each method has its advantages depending on the specific equation. Mastery of these techniques is essential for finding the roots of the equations derived from the intersection of the two functions.