In Exercises 115–122, find all values of x satisfying the given conditions. y = 5x^2 + 3x and y = 2

Verified step by step guidance

1. Start by setting the two expressions for y equal to each other: $5x^2+3x=2$.
2. Rearrange the equation to form a standard quadratic equation: $5x^2+3x-2=0$.
3. Identify the coefficients for the quadratic formula: $a=5$, $b=3$, and $c=-2$.
4. Use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ to find the values of $x$.
5. Calculate the discriminant $b^2-4ac$ to determine the nature of the roots.

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 = 5x^2 + 3x represents a parabola that opens upwards since the coefficient of x^2 (which is 5) is positive. Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and roots, 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 accomplished through various methods, including factoring, completing the square, or using the quadratic formula. In this scenario, after setting the equations equal, we will likely rearrange the equation into standard form (ax^2 + bx + c = 0) and apply one of these methods to find the roots. Understanding these techniques is crucial for effectively solving the problem.