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Word Problems: Quadratic Equations

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Word Problems: Quadratic Equations

Study Guide

Key Definition

A quadratic equation is an equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a \neq 0$.

Important Notes

Quadratic equations are used to find unknown lengths in geometry.
The Pythagorean Theorem is often applied in these problems: $a^2 + b^2 = c^2$.
Factoring is a common method to solve quadratic equations.
Completing the square and the quadratic formula are alternative methods.
In $x^2$, the exponent 2 indicates the power to which x is raised.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$^$ represents exponentiation

$√$ represents square root

$±$ represents plus or minus

Remember to use proper notation when solving problems

Why It Works

Quadratic equations allow us to model and solve real-world problems involving areas, velocities, and other quantities that vary quadratically.

Remember

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Quick Reference

Pythagorean Theorem:$a^2 + b^2 = c^2$

Standard Form:$ax^2 + bx + c = 0$

Understanding Word Problems: Quadratic Equations

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A quadratic word problem translates a real scenario into an equation with an $x^2$ term. For example: "A rectangle has sides x meters and (x + 2) meters, with area 15 m²," gives $x(x + 2) = 15$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Solve for $x$ in $x^2 - 4x - 12 = 0$.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have a right-angled triangular garden with sides that are three consecutive even numbers (in centimeters). Use the Pythagorean theorem to find the lengths.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a scenario where the area of a rectangle in square meters is represented by $x^2 + 5x + 6$. What are the possible dimensions of the rectangle in meters?

Challenge Quiz

Single Choice Quiz

Advanced

Solve the quadratic equation $2x^2 - 4x - 6 = 0$.

Recap

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Review key concepts and takeaways