Factoring Quadratic Equations

Study Notes

A quadratic equation in the form of ax^2 + bx + c = 0 can be factored into (x - r)(x - s) = 0, where r and s are the roots of the equation.
Factoring is possible when the equation can be written in the form of a product of two binomials.
Steps to factor a quadratic equation:
1. Look for two numbers whose product is c (the constant term) and whose sum is b (the coefficient of the x term).
2. Rewrite the middle term as the sum of these two numbers.
3. Factor by grouping.

Quadratic Formula: 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 will always give two solutions for the value of x.
Graphical Method: Quadratic equations can be solved graphically by finding the x-intercepts of the corresponding quadratic curve.
Completing the Square: A method of solving quadratic equations by rearranging the equation to have a perfect square trinomial on one side and zero on the other side.

Projectile Motion: Quadratic equations are used to model the trajectory of projectiles under gravity, such as the trajectory of a thrown ball.
Optimization Problems: Quadratic equations are used to solve optimization problems, such as finding the maximum or minimum value of a quadratic function.
Electrical Circuits: Quadratic equations are used to analyze and design electrical circuits, such as filters and amplifiers.
Physics and Engineering: Quadratic equations are used to model and solve problems in physics and engineering, such as the motion of objects, force, and energy.

A quadratic equation in the form of ax^2 + bx + c = 0 can be factored into (x - r)(x - s) = 0, where r and s are the roots of the equation.
Factoring is possible when the equation can be written in the form of a product of two binomials.
Steps to factor a quadratic equation include looking for two numbers whose product is c and whose sum is b, rewriting the middle term as the sum of these two numbers, and factoring by grouping.

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 always gives two solutions for the value of x.
The Graphical Method involves finding the x-intercepts of the corresponding quadratic curve.
Completing the Square is a method of solving quadratic equations by rearranging the equation to have a perfect square trinomial on one side and zero on the other side.

Quadratic equations are used to model the trajectory of projectiles under gravity, such as the trajectory of a thrown ball.
Quadratic equations are used to solve optimization problems, such as finding the maximum or minimum value of a quadratic function.
Quadratic equations are used to analyze and design electrical circuits, such as filters and amplifiers.
Quadratic equations are used to model and solve problems in physics and engineering, such as the motion of objects, force, and energy.