Quadratic Equations and Inequalities

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.