ACT Math
A comprehensive course covering the essential math concepts and strategies needed to excel on the ACT.
Advanced Topics
Quadratic Equations and Polynomials
Parabolas and More
Quadratic equations have the form \( ax^2 + bx + c = 0 \). Their graphs are U-shaped curves called parabolas.
Solving Quadratics
- Factoring: Break into two binomials.
- Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
- Completing the Square: Rearranging into a perfect square.
Polynomials
- An expression with several terms, like \( 3x^3 + 2x^2 - x + 5 \).
Real-World Application
Quadratics are used in physics (projectile motion), engineering, and finance (calculating profit).
ACT Focus
Recognize graphs, solve for roots, and interpret word problems involving quadratics.
Key Formula
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Examples
Solving \( x^2 - 5x + 6 = 0 \) by factoring: \( (x-2)(x-3) = 0 \), so \( x = 2 \) or \( x = 3 \)
Using the quadratic formula on \( x^2 + 2x - 8 = 0 \) gives \( x = 2 \) or \( x = -4 \)
In a Nutshell
Quadratics pop up whenever something curves—like paths of thrown balls or profit calculations.