Solving Quadratic Equations: Methods and Applications

Quadratic Equations

Standard Form and Key Properties

Quadratic equations are polynomial equations of degree two, typically written in the standard form:

• Standard Form: $ax^2 + bx + c = 0$

• Square Root Property: If $x^2 = k$, then $x = rac{1}{2} ext{ or } x = - rac{1}{2}$

Quadratic equations can be solved using several methods, including factoring, completing the square, and the quadratic formula.

Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation:

• Formula: $x = rac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$

• Discriminant: The expression under the square root, $b^2 - 4ac$, determines the nature of the solutions (real and distinct, real and equal, or complex).

Methods for Solving Quadratic Equations

Factoring

Factoring is used when the quadratic can be written as a product of two binomials:

• Set the equation to zero: $ax^2 + bx + c = 0$

• Factor into $(x + p)(x + q) = 0$

• Set each factor equal to zero and solve for $x$

Example: $x^2 - x - 22 = 0$ $(x - 5)(x + 4) = 0$ Solution set: $x = 5, x = -4$

Completing the Square

Completing the square transforms the quadratic into a perfect square trinomial:

• Move the constant to the other side: $x^2 + bx = c$

• Add $( rac{b}{2})^2$ to both sides

• Rewrite as $(x + rac{b}{2})^2 = d$

• Take the square root of both sides and solve for $x$

Example: $x^2 - 4x - 8 = 0$ Add $4$to both sides:$x^2 - 4x + 4 = 12(x - 2)^2 = 12x - 2 = \\pm \\sqrt{12}x = 2 \\pm 2\\\sqrt{3}$

Using the Quadratic Formula

Apply the quadratic formula when factoring is not straightforward:

• Identify $a$, $b$, and $c$ from $ax^2 + bx + c = 0$

• Substitute into the formula $x = rac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$

• Simplify to find the solution set

Example: $y^2 + 14y = 30$ $y^2 + 14y - 30 = 0$ $a = 1, b = 14, c = -30$ $y = rac{-14 \\pm \\sqrt{14^2 - 4(1)(-30)}}{2}$ $y = rac{-14 \\pm \\sqrt{196 + 120}}{2}$ $y = rac{-14 \\pm \\sqrt{316}}{2}$ $y = -7 \\pm \\sqrt{79}$

Special Cases and Applications

Zero-Factor Property

If a quadratic equation can be factored such that one factor equals zero, the solutions are the roots of each factor.

Example: $(x + 1)(x - 2) = 0$ Solution set: $x = -1, x = 2$

Square Root Property

Used when the equation is in the form $x^2 = k$:

• Take the square root of both sides: $x = \\pm \\sqrt{k}$

Example: $y^2 = 49$ $y = \\pm 7$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Steps
Factoring	When the quadratic factors easily	Set to zero, factor, solve each factor
Completing the Square	When $a = 1$ or easily made $1$	Isolate terms, add $(b/2)^2$, solve
Quadratic Formula	Always applicable	Identify $a$, $b$, $c$, substitute, solve
Square Root Property	When in $x^2 = k$ form	Take square root, solve

Additional info:

• Examples in the notes cover all major solution methods for quadratic equations, including cases with irrational and complex solutions.

• Each method is illustrated with step-by-step solutions, reinforcing procedural understanding.