Find all values of x satisfying the given conditions. y₁ = - x² + 4x - 2, y₂ = - 3x² + x - 1, 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: (-x^2 + 4x - 2) - (-3x^2 + x - 1) = 0.
2. Step 2: Simplify the equation by distributing the negative sign for y2 and combining like terms. This results in: -x^2 + 4x - 2 + 3x^2 - x + 1 = 0.
3. Step 3: Combine the x^2 terms, x terms, and constant terms. This simplifies to: (3x^2 - x^2) + (4x - x) + (-2 + 1) = 0, which becomes 2x^2 + 3x - 1 = 0.
4. Step 4: Solve the quadratic equation 2x^2 + 3x - 1 = 0 using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a = 2, b = 3, and c = -1.
5. Step 5: Substitute the values of a, b, and c into the quadratic formula and simplify under the square root (discriminant) and the fraction to find the two possible solutions for x.

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.

Difference of Functions

The difference of two functions, represented as y1 - y2, is a new function that combines the outputs of y1 and y2. In this problem, we are tasked with finding where this difference equals zero, which indicates the points of intersection. Analyzing the resulting quadratic equation from this difference is crucial for determining the values of x that satisfy the given conditions.