Quadratic Equations, Functions, Zeros, and Models

Standard Form of a Quadratic Equation

A quadratic equation is an equation that can be written in the form:

where and . This is called the standard form of a quadratic equation.

Zero-Factor Property

The Zero-Factor Property states that if , then or (or both). This property is fundamental for solving quadratic equations by factoring.

• Example: Solve .

• Solution: Rearranging, . Factor if possible, then set each factor equal to zero and solve for .

Square Root Property

The Square Root Property is used to solve equations of the form :

• If , then or , or more concisely, .

• Example: Solve .

• Solution: .

Solving Quadratic Equations by Completing the Square

Completing the square is a method for solving quadratic equations by rewriting the equation so that one side is a perfect square trinomial.

• Steps:

  1. If , divide both sides by .

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

  3. Take half the coefficient of , square it, and add to both sides.

  4. Write the left side as a squared binomial.

  5. Use the square root property to solve for .

• Example: Solve by completing the square.

  • Move constant:

  • Half of 10 is 5;

  • Add 25 to both sides:

  • Left side factors:

  • Take square root:

  • Solve:

Quadratic Formula

The Quadratic Formula provides the solutions to any quadratic equation :

• Example: Solve .

  • Rewrite:

  • Here, , ,

  • Plug into formula:

Zeros of a Quadratic Function

The zeros of a quadratic function are the values of for which .

• Example: Find the zeros of .

• Set and solve using the quadratic formula.

The Discriminant

The discriminant is the expression under the square root in the quadratic formula: . It determines the nature and number of solutions:

• Example: For , discriminant is (one real double root).

Equations Quadratic in Form

An equation is quadratic in form if it can be written as , where is an algebraic expression.

• Example: Solve .

  • Let , so equation becomes

  • Factor:

  • So or

  • Substitute back: ;

Applications of Quadratic Equations

Quadratic equations are commonly used to model real-world problems, such as geometry and physics.

• Example: One leg of a right triangle is 7 cm less than the other leg. The hypotenuse is 13 cm. Find the lengths of the legs.

  • Let = length of one leg; other leg =

  • By Pythagoras:

  • Expand:

  • Combine:

  • Subtract 169:

  • Divide by 2:

  • Factor:

  • So (since length must be positive), other leg =

Additional info: The notes above include expanded explanations, step-by-step examples, and a summary table for the discriminant, as would be found in a modern college algebra textbook.