Quadratic Equations in One Variable

Definition and Standard Form

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

• a, b, and c are real numbers with .

• This is called the standard form of a quadratic equation.

• Quadratic equations are also known as second-degree equations because the highest power of the variable is 2.

Examples of quadratic equations:

• (can be rewritten as )

Solving a Quadratic Equation

Overview of Solution Methods

Quadratic equations can have up to two solutions because they are second-degree equations. There are four main methods to solve them:

1. Factoring (Zero-Factor Property)

2. Square-root property

3. Completing the square

4. Quadratic formula

Zero-Factor Property

The zero-factor property states: If , then either , , or both.

• This property is used when a quadratic equation can be factored into two binomials.

• Factoring is often the fastest and easiest way to solve a quadratic equation if possible.

Steps for Solving by Factoring:

1. Get all terms on one side of the equation (so the other side is 0).

2. Arrange terms in order: , , constant.

3. Factor the quadratic expression.

4. Set each factor equal to zero and solve for .

Example: Solve

• Rewrite:

• Factor:

• Set each factor to zero: or

• Solutions: or

Square-Root Property

The square-root property is used when the quadratic equation can be written in the form .

If , then

• This method is best when there is no -term or when the equation is a perfect square.

• We do not show radicals on both sides; simply apply the square root to both sides.

Example: Solve

• Take the square root:

Example: Solve

• Take the square root:

• Solve for :

Completing the Square

Completing the square is a method used to solve any quadratic equation by rewriting it as a perfect square trinomial.

Steps for Completing the Square:

1. If , divide both sides by .

2. Rewrite so the constant term is alone on one side.

3. Divide the coefficient of by 2 and square it. Add this value to both sides.

4. Factor the left side as a perfect square; combine like terms on the right.

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

Example: Solve

• Rewrite:

• Take half of (), square it (), add to both sides:

• Left side factors:

• Take square root:

• Solution:

Quadratic Formula

The quadratic formula can solve any quadratic equation of the form :

• Use this formula if factoring and square-root property are not possible.

• Always substitute values for , , and directly into the formula.

• Remember: the fraction bar extends under the entire numerator, including the and the radical.

Example: Solve

• Rewrite:

• , ,

• Plug into formula:

The Discriminant

Definition and Use

The discriminant is the expression under the radical in the quadratic formula:

• The discriminant tells us the nature and number of solutions:

If the discriminant is a perfect square, the solutions are rational and factoring is possible.

Choosing the Best Method to Solve Quadratic Equations

• If there is no -term or if the equation is a squared binomial, use the square-root property.

• If factoring is possible, use the zero-factor property.

• If neither factoring nor the square-root property is possible, use the quadratic formula.

Summary Table: Methods for Solving Quadratic Equations

Practice Problems

• Solve by completing the square.

• Solve by factoring or formula.

• Solve using the square-root property.

• Solve using the quadratic formula.

Additional info: The notes emphasize the importance of understanding all four methods, as each is useful in different situations. Mastery of the quadratic formula is especially important for exam preparation.