Find all values of x satisfying the given conditions. y₁ = 2x² + 5x - 4, y₂ = - x² + 15x - 10, and y₁ - y₂ = 0

Verified step by step guidance

1. Step 1: Start by substituting the given expressions for y1 and y2 into the equation y1 - y2 = 0. This gives (2x^2 + 5x - 4) - (-x^2 + 15x - 10) = 0.
2. Step 2: Simplify the equation by distributing the negative sign across the terms in y2. This results in 2x^2 + 5x - 4 + x^2 - 15x + 10 = 0.
3. Step 3: Combine like terms. Group the x^2 terms, the x terms, and the constant terms together. This simplifies to 3x^2 - 10x + 6 = 0.
4. Step 4: Factorize the quadratic equation 3x^2 - 10x + 6 = 0, if possible. Alternatively, use the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a, where a = 3, b = -10, and c = 6.
5. Step 5: Solve for the two possible values of x by substituting the values of a, b, and c into the quadratic formula or by solving the factored form of the equation. These values of x are the solutions to the problem.

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.