Solving Quadratic Equations: Square Root Property and Factoring

Quadratic Equations

Introduction to Solving Quadratic Equations

Quadratic equations are equations of the form ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Not all quadratic equations can be solved by factoring; therefore, alternative methods are necessary.

• Factoring is possible only when the quadratic can be written as a product of two binomials.

• There are other methods to solve quadratics, such as the square root property, completing the square, and the quadratic formula.

Methods for Solving Quadratic Equations

There are four main methods for solving quadratic equations:

Method	When to Use	Steps
Factoring	When the equation can be factored into binomials	1. Write in standard form 2. Factor completely 3. Set each factor to zero 4. Solve for x 5. (Optional) Check solutions
Square Root Property	When the equation is in the form $x^2 = k$ or $(ax + b)^2 = k$	1. Isolate the squared expression 2. Take the square root of both sides 3. Solve for x 4. (Optional) Check solutions
Method #3	Additional methods (e.g., completing the square)	See textbook for details
Method #4	Quadratic formula	See textbook for details

Factoring Quadratic Equations

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

• $x^2 + x - 6 = 0$ can be factored as $(x + 3)(x - 2) = 0$

• $x^2 - 5 = 0$ cannot be factored over the integers

Factorable? If yes, use factoring. If no, use another method.

Square Root Property

The square root property is used when the quadratic equation is in the form $x^2 = k$ or $(ax + b)^2 = k$. The property states:

• If $x^2 = k$, then $x = oxed{ ext{±} \, ext{sqrt}(k) }$

• If $(ax + b)^2 = k$, then $ax + b = ext{±} \, ext{sqrt}(k)$

Steps for Using the Square Root Property:

1. Isolate the squared expression

2. Take the square root of both sides

3. Solve for the variable

4. (Optional) Check solutions

Example 1: Solving Using the Square Root Property

• Given: $(x + 1)^2 = 4$

• Solution:

  • Take the square root: $x + 1 = ext{±}2$

  • Solve for x: $x = 1$ or $x = -3$

Example 2: Solving with a Coefficient

• Given: $4x^2 - 5 = 0$

• Solution:

  • Isolate $x^2$: $4x^2 = 5$

  • Divide by 4: $x^2 = rac{5}{4}$

  • Take the square root: $x = ext{±} rac{ ext{sqrt}(5)}{2}$

Practice Problems

• Given: $2x^2 - 16 = 0$

• Solution:

  • Isolate $x^2$: $2x^2 = 16$

  • Divide by 2: $x^2 = 8$

  • Take the square root: $x = ext{±} ext{sqrt}(8)$

  • Simplify: $x = ext{±} 2 ext{sqrt}(2)$

Imaginary Roots

Complex Solutions in Quadratic Equations

Sometimes, solving a quadratic equation using the square root property results in imaginary (complex) roots. This occurs when the value under the square root is negative.

• Imaginary roots are written using the imaginary unit $i$, where $i = ext{sqrt}(-1)$.

• If $x^2 = -k$, then $x = ext{±} ext{sqrt}(-k) = ext{±} i ext{sqrt}(k)$

Example: Imaginary Roots

• Given: $4x^2 + 25 = 0$

• Solution:

  • Isolate $x^2$: $4x^2 = -25$

  • Divide by 4: $x^2 = - rac{25}{4}$

  • Take the square root: $x = ext{±} rac{5i}{2}$

Note: When both a and c have the same sign in standard form, you will always end up with a complex answer.

Summary Table: Square Root Property Steps

Step	Description
1	Isolate squared expression
2	Take ± square root
3	Solve for variable
4 (Optional)	Check solutions

Additional info: The notes reference "function" and "radical" as possible types of solutions, which connects to broader topics in College Algebra such as function notation and radical expressions.