Illustrations of Quadratic Equations

Study Notes

Pigs and Chickens Problem

Farmer has a total of 78 feet and 27 heads of pigs and chickens.
Let x represent pigs and y represent chickens.
Formulated equations:
- Eq. 1: (4x + 2y = 78) (Feet)
- Eq. 2: (x + y = 27) (Heads)
Solved to find:
- (x = 12) (pigs)
- (y = 15) (chickens)

Linear and Quadratic Equations

Linear Equation: A mathematical sentence of degree 1, standard form (ax + by = c).
Quadratic Equation: A mathematical sentence of degree 2, standard form (ax^2 + bx + c = 0).

Examples of Quadratic Equations

Determined various expressions:
- (3x^2 + 5x - 3 = 0) is quadratic.
- (5m^5 + 2m^2 - 6 = 0) is not quadratic.

Nature of Roots

Discriminant ((b^2 - 4ac)):
- (= 0): Real and equal roots
- (> 0): Rational and unequal roots
- (< 0): No real roots

Completing the Square

Objective: Solve quadratic equations by transforming them into perfect squares.
Steps involve rewriting equations and determining necessary terms to add for completion.

Solving Quadratics by Factoring

Involves finding roots by factorization of quadratic equations.
Use example patterns of ( (x + p)^2 = q ).

Perfect Squares and Their Root Values

Perfect squares of integers from 1 to 30 are listed from (1^2 = 1) to (30^2 = 900).
Not perfect squares include specific values with their square roots provided.

Sum and Product of the Roots

Defined relationships for roots of quadratic equations:
- Sum of the roots: (-b/a)
- Product of the roots: (c/a)
Example solved for (x^2 - 12x + 20) yields:
- Sum = 12
- Product = 20

Quadratic Formula

General solution model: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
Discriminant's role in determining nature of roots reflected through various examples.

Example Evaluations

Calculated discriminants providing insights into root types:
- (b=7, c=10) produces rational roots.
- (b=2, c=5) produces no real roots.

Summary

Quadratic equations dominate as mathematical tools for various applications including problem-solving in real-world scenarios.
Understanding forms, roots, and their conditions are critical for mastery in algebra.

Description

Explore the Illustrations of Quadratic Equations through the Pigs and Chickens Problem. This quiz examines a real-life scenario involving algebraic equations to determine the number of pigs and chickens on a farm. Test your understanding of quadratic relationships and problem-solving skills.