Quadratic Equations and Inequalities

Study Notes

Solving Quadratic Inequalities

A quadratic inequality is an inequality of the form ax^2 + bx + c > 0, ax^2 + bx + c ≥ 0, ax^2 + bx + c < 0, or ax^2 + bx + c ≤ 0, where a, b, and c are real numbers and a ≠ 0.
To solve a quadratic inequality, follow these steps:
1. Write the inequality in standard form (ax^2 + bx + c > 0, etc.).
2. Factor the left-hand side, if possible.
3. Find the critical points (roots) by setting the factored expressions equal to 0 and solving for x.
4. Plot the critical points on a number line.
5. Test a point in each region of the number line to determine the solution.

Applications Of Quadratic Equations

Projectile motion: Quadratic equations can be used to model the trajectory of projectiles, such as the height of a thrown ball or the range of a projectile.
Optimization problems: Quadratic equations can be used to find the maximum or minimum value of a function, such as the maximum area of a rectangle with a fixed perimeter.
Physics and engineering: Quadratic equations are used to model the motion of objects, including the acceleration and velocity of particles and the vibration of springs.

Quadratic Formula

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
The quadratic formula can be used to solve any quadratic equation, whether it can be factored or not.
The ± symbol indicates that there may be two solutions to the equation.

Graphing Quadratic Functions

The graph of a quadratic function is a parabola that opens upward or downward.
The vertex of the parabola is the minimum or maximum point of the function.
The x-intercepts of the parabola are the solutions to the quadratic equation.
The graph can be used to identify the axis of symmetry, the vertex, and the x-intercepts.

Factoring Quadratic Expressions

Factoring a quadratic expression involves expressing it as a product of two binomials: (x + r)(x + s) = x^2 + (r + s)x + rs.
The factors of a quadratic expression can be found by looking for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Factoring can be used to solve quadratic equations, as the solutions are the values of x that make each factor equal to 0.

Description

Test your understanding of quadratic equations, inequalities, and functions. Learn how to solve quadratic inequalities, apply quadratic equations to real-world problems, and graph quadratic functions. Practice factoring quadratic expressions and using the quadratic formula.