Algebra

Fundamental algebraic concepts including equations, inequalities, and functions.

Variables and ExpressionsEquations and SolutionsInequalities
Functions and GraphsSystems of EquationsQuadratic Equations
Budgeting and Saving MoneyTravel Planning with EquationsCooking and Recipes
Checking Your WorkBreaking Problems Into Steps
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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.

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