Algebra
Fundamental algebraic concepts including equations, inequalities, and functions.
Advanced Topics
Quadratic Equations
What is a Quadratic Equation?
A quadratic equation has the form \( ax^2 + bx + c = 0 \), where \( a eq 0 \). The highest power of \( x \) is 2.
How to Solve Quadratics
- Factoring: Find two numbers that multiply to \( c \) and add to \( b \).
- Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
- Completing the Square: Rearrange the equation to create a perfect square.
Graphs of Quadratics
The graph is a parabola, which looks like a "U" shape. The vertex is the highest or lowest point.
Real-World Quadratics
Quadratic equations are used in physics, engineering, and predicting how objects move, like a basketball shot or a rocket's path.
Key Formula
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Examples
Solve \( x^2 - 4 = 0 \): \( x^2 = 4 \), so \( x = 2 \) or \( x = -2 \).
The graph of \( y = x^2 - 2x + 1 \) is a parabola opening upward.
In a Nutshell
Quadratic equations involve \( x^2 \) and their graphs are parabolas.