Quadratic Equations and Their Roots

Study Notes

Quadratic Formula

The formula is defined as x = -b ± √(b² - 4ac)/2a.
This formula provides the solutions for any quadratic equation of the form ax² + bx + c = 0.
Always substitute the respective values of "a," "b," and "c" into the formula for solving.

Nature of the Roots

Refers to the type of roots present in a quadratic equation: can be real, rational, or imaginary.
Determined by the discriminant (b² - 4ac); its value influences the nature of roots.

Irrational vs Rational

Irrational roots arise from non-perfect square discriminants (b² - 4ac < 0).
Rational roots occur when the discriminant is a perfect square (b² - 4ac = 0).

Location of Roots

Roots are found where the quadratic function equals zero (y = 0).
Graphically represented as the points where the parabola crosses the x-axis.

Solving using Square Roots

Involves isolating the x² term and taking the square root of both sides.
Remember to include both the positive and negative solutions (±).

Equal vs Unequal

Solutions that equal zero indicate the roots of the quadratic function.
If not equal to zero, it reflects that the function does not intersect the x-axis.

Real vs Imaginary

Real roots are positive or negative numbers.
Imaginary roots are complex numbers that result from a negative discriminant.

CTS Method (Completing the Square)

A method to solve quadratic equations by rewriting them in the form (x - p)² = q.
Involves taking half of the "b" coefficient, squaring it, and adding it to complete the square.

Description

This quiz covers the Quadratic Formula and its implications in solving quadratic equations. You'll learn how to identify the nature of roots based on the discriminant and differentiate between rational and irrational roots. Additionally, it explores the graphing of quadratic functions and locating roots on a parabola.