In Exercises 115–122, find all values of x satisfying the given conditions. y1 = - x^2 + 4x - 2, y2 = - 3x^2 + x - 1, and y1 - y2 = 0

Verified step by step guidance

1. Start by setting the equation y1 - y2 = 0, which means -x^2 + 4x - 2 - (-3x^2 + x - 1) = 0.
2. Simplify the equation by distributing the negative sign: -x^2 + 4x - 2 + 3x^2 - x + 1 = 0.
3. Combine like terms: (3x^2 - x^2) + (4x - x) + (-2 + 1) = 0.
4. This simplifies to 2x^2 + 3x - 1 = 0.
5. Solve the quadratic equation 2x^2 + 3x - 1 = 0 using the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a = 2, b = 3, and c = -1.

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, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the properties of these functions, such as their vertex, axis of symmetry, and roots, is essential for solving equations involving them.

Finding Intersections

Finding the intersection 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 requires rearranging the equation into standard form and applying methods such as factoring, completing the square, or using the quadratic formula.

Zeroes of a Function

The zeroes of a function are the values of x that make the function equal to zero. For quadratic functions, these can be found using various methods, including factoring, the quadratic formula, or graphing. Identifying the zeroes is crucial for understanding the behavior of the function and for solving equations where the function equals another expression.