Math unit 3 on 10 March 2025 at 12.07.12 PM

Quadratic Equations Overview

Quadratic equations are mathematical expressions of the standard form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants and ( a
eq 0 ). This formula represents a parabola when graphed. To transform an equation into standard form, one must manipulate it so that all terms are on one side and the equation equals zero. Understand that coefficients are the numerical factors in front of their respective variable terms, significantly influencing the shape and position of the parabola on the coordinate plane.

Solving Quadratics by Factoring

• Example: To solve an equation where the coefficient of ( x^2 ) is one, like ( x^2 + bx + c = 0 ), identify two factors of the constant term ( c ) that sum up to the coefficient of the linear term ( b ).

• For example, given the quadratic equation ( x^2 - 7x + 12 = 0 ), we look for factors of 12 that yield negative sums, which are (-3) and (-4). The expression can then be factored as ( (x - 3)(x - 4) = 0 ).

Using a Graphing Calculator:

To find zeros or roots of the quadratic function, utilize the second trace function and select the 'zero' option. During this process, it's essential to note that the exact values of the intersections may be fractions. However, approximate solutions like ( x = 3 ) and ( x = 4 ) can typically be found using graphing techniques.

Finding Factors:

Understanding the relationship between zeros and factors is crucial. If ( x = r ) is a zero, then the corresponding factor can be expressed as ( (x - r) ). Following our example, if ( x = 3 ) is a zero, the corresponding factor is ( (x - 3) ), confirming our earlier factorization: ( (x - 3)(x - 4) = 0 ).

Factoring By Grouping Technique

• Example breakdown: Consider the equation ( 9x^2 - 15x = 0 ). First, factor out the greatest common factor (GCF), it simplifies our work and can often yield negative solutions.

• Grouping involves recognizing common factors in different pairs and factoring them out accordingly, ensuring that factors yield the original equation: ( (5x - 3)(x - 1) = 0 ).

Methods of Solving Quadratics

1. Factoring: The process of breaking down the quadratic into manageable products which can be solved.

2. Quadratic Formula: The quadratic formula, ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), provides solutions for any quadratic equation, accommodating for all scenarios.

3. Square Root Property: When ( x^2 ) is isolated, solutions can often be derived directly. For instance, the equation ( 3x^2 = 9 ) simplifies to ( x = \pm\sqrt{3} ).

Understanding Complex Solutions

The discriminant (the part of the quadratic formula noted as ( b^2 - 4ac )) dictates the nature of the solutions:

• If the discriminant is greater than 0, the equation has two distinct real solutions.

• If it equals 0, the parabola touches the x-axis at one point, indicating one real solution.

• If it is less than 0, there are no real solutions, leading to two complex solutions, illustrated graphically by parabolas that do not intersect the x-axis.

Completing the Square Method

Steps to completing the square:

1. Move the constant term to the other side of the equation.

2. Add ( (\frac{b}{2})^2 ) to both sides to form a perfect square trinomial.

3. Solve for ( x ). For example, transforming the equation ( x^2 - 6x + 9 = ??? ) leads to ( (x - 3)^2 = 19 ). This alternate method reveals potential solutions more easily with the appropriate insight.

Introduction to Complex Numbers

• Complex Numbers: These numbers contain real parts and imaginary units (denoted as ( i )), where ( i ) is defined as the square root of -1. For example, the square root of -4 would transform into 2i.

Operations with Complex Numbers

• Addition/Subtraction: To perform these operations, combine the real parts and imaginary parts separately. For example, ( (2 + 3i) - (4 + 5i) = -2 - 2i ).

• Multiplication: Distributing terms often involves simplifying ( i^2 ) to reflect negative values. Therefore, when multiplying ( (a + bi)(a - bi) ), the result is ( a^2 + b^2 ).

Simplifying Complex Fractions

To eliminate the imaginary unit in the denominator during division, use the complex conjugate. For instance, simplifying ( \frac{2 + 3i}{5 - 6i} ) entails multiplying both the numerator and the denominator by ( (5 + 6i) ). This will cause certain components to cancel out, resulting in simplified quadratic expressions.

Summary

Various robust methods exist for solving quadratic equations, including factoring, completing the square, and utilizing the quadratic formula. A comprehensive understanding of complex numbers and their operations enhances solving capabilities, particularly for cases where real solutions are absent. To master these concepts, this course will cover multiple sections on quadratic equations to ensure a thorough understanding of the fundamental principles underpinning their nature and applications.