Find all values of x satisfying the given conditions. y = 2x² - 3x and y = 2

Verified step by step guidance

1. Step 1: Start by setting the two equations equal to each other since both represent y. This gives the equation 2x^2 - 3x = 2.
2. Step 2: Rearrange the equation to set it equal to 0. Subtract 2 from both sides to get 2x^2 - 3x - 2 = 0.
3. Step 3: Factor the quadratic equation, if possible. Look for two numbers that multiply to the product of the leading coefficient (2) and the constant term (-2), and add to the middle coefficient (-3). Alternatively, use the quadratic formula if factoring is not straightforward.
4. Step 4: Solve for x using the factors or the quadratic formula. If using the quadratic formula, apply x = (-b ± √(b^2 - 4ac)) / (2a), where a = 2, b = -3, and c = -2.
5. Step 5: Verify the solutions by substituting the x-values back into the original equations to ensure they satisfy both y = 2x^2 - 3x and y = 2.

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 represents a parabola that opens upwards, as the coefficient of x^2 (which is 2) is positive. Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and intercepts, 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 graphs. This involves setting the two equations equal to each other, which allows us to solve for x. The solutions represent the x-coordinates where the parabola intersects the horizontal line y = 2, providing the required values of x.

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, once we set the equations equal to each other, we will likely rearrange the equation into standard form (ax^2 + bx + c = 0) and then apply one of these methods to find the values of x that satisfy the equation.