Find all values of x satisfying the given conditions. y = 5x² + 3x and y = 2

Verified step by step guidance

1. Start by setting the two equations equal to each other since both represent y. This gives: $5x^2\; +\; 3x\; =\; 2$.
2. Rearrange the equation to set it equal to 0, which is the standard form for a quadratic equation: $5x^2\; +\; 3x\; -\; 2\; =\; 0$.
3. Identify the coefficients of the quadratic equation: $a\; =\; 5$, $b\; =\; 3$, and $c\; =\; -2$. These will be used in the quadratic formula.
4. Apply the quadratic formula: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$. Substitute the values of $a$, $b$, and $c$ into the formula.
5. Simplify the discriminant $b^2\; -\; 4ac$ and then compute the two possible solutions for $x$ by evaluating the formula with both the plus and minus options for the square root.

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, where 'a' 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

Finding the intersection of two functions involves determining the values of x where the functions are equal. In this problem, we set the quadratic function equal to the constant function y = 2 to find the x-values that satisfy both equations. This process often requires rearranging the equation into standard form and applying methods such as factoring, completing the square, or using the quadratic formula.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. It provides the solutions for x based on the coefficients a, b, and c. In this context, once the equation is rearranged to standard form, applying the quadratic formula will yield the x-values where the two functions intersect.