Quadratic Equations

Definition and Standard Form

A quadratic equation in one variable x is any equation equivalent to the form:

• Standard form: , where and .

• Also called a second-degree equation.

Examples:

• (here, , , )

• is equivalent to (, , )

• (, , )

Solving Quadratic Equations by Factoring

Factoring is effective when the quadratic can be written as a product of linear factors. The Zero-Product Property states: if , then or .

• Write the equation in standard form ().

• Factor the quadratic expression.

• Set each factor equal to zero and solve for .

Examples:

• or Solution set:

• or Solution set:

• or Solution set:

• Factor as or Solution set:

Solving Quadratic Equations by the Square-Root Property

The Square-Root Property states: If (with ), then .

• Applies to equations of the form or .

• Remember: means two solutions: and .

Examples:

• Solution set:

• or Solution set:

Solving Quadratic Equations by Completing the Square

Completing the square is a method to rewrite a quadratic equation in the form .

1. Move the constant term to the other side: .

2. If , divide both sides by .

3. Add to both sides.

4. Write the left side as a perfect square: .

5. Use the square-root property to solve for .

Examples:

• Add to both sides: or Solution set:

• Add to both sides: Solution set:

Solving Quadratic Equations by the Quadratic Formula

The Quadratic Formula solves any quadratic equation :

• Always works, even when factoring is not possible.

Example:

• , , Solution set:

The Discriminant and Nature of Solutions

The discriminant of a quadratic equation is .

• If : two distinct real solutions.

• If : one real solution (a repeated root).

• If : no real solutions (but two complex solutions).

Example:

• Rearranged: Since , no real solutions.

Applications of Quadratic Equations

• Area Problems: If a rectangle's length is 9 yards more than its width and area is 400 sq. yards: Let = width, (since width must be positive), Width: 16 yards, Length: 25 yards

• Pythagorean Problems: TV screen with width 12.2 inches more than height, diagonal 32 inches. Let = height, , Use quadratic formula to solve for $h$ (positive root only): ,

Complex Numbers and Quadratic Equations in the Complex Number System

The Set of Complex Numbers

• The imaginary unit is defined by .

• The set of complex numbers :

• Standard form:

• Real part: ; Imaginary part:

• Pure imaginary: (e.g., )

• Every real number is a complex number ()

• Modulus:

Arithmetic with Complex Numbers

Addition and Subtraction

Examples:

Multiplication, Conjugation, and Division

• Complex conjugate:

• Product with conjugate:

• Reciprocal:

Examples:

Powers of i

The pattern repeats every four powers.

Examples:

Solving Quadratic Equations with Complex Numbers

• If the discriminant , solutions are complex conjugates.

• Principal square root of a negative: for

Examples:

• Solution set:

• Solution set:

Radical Equations, Equations of Quadratic Type, and Factorable Equations

Radical Equations

A radical equation is one in which the variable appears under a radical sign. To solve:

1. Isolate the radical.

2. Raise both sides to the power equal to the index of the radical.

3. Solve the resulting equation.

4. Check all solutions for extraneous roots.

Examples:

• Cube both sides: Solution set:

• Square both sides: or Check: Only is valid. Solution set:

• Isolate and square both sides twice, solve resulting quadratic, check solutions. Solution set:

Equations of Quadratic Form

Some equations are not quadratic in , but can be made quadratic by substitution.

Examples:

• Let , so or Back-substitute: ; Solution set:

• Let , so or ; Solution set:

Solving Equations by Factoring

Some higher-degree equations can be solved by factoring.

Examples:

• Solution set:

• Factor by grouping: Solution set:

Summary Table: Nature of Solutions of Quadratic Equations