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Learn factoring quadratic equations into (x-p)(x-q) form. Learn systematic steps, grouping methods, and special cases with worked examples and practice problems.
---
## What is Quadratic Equation Factorization?
Quadratic equation factorization is the process of converting an equation from the form $$ax^2 + bx + c = 0$$ to the form $$a(x - p)(x - q) = 0$$, where $$p$$ and $$q$$ are the roots of the quadratic equation.
Visible text: Quadratic equation factorization is the process of converting an equation from the form to the form , where and are the roots of the quadratic equation.
Note that the roots of a quadratic equation are the values of $$x$$ that make the equation equal to zero. When we convert the equation to its factored form, we can easily find its roots.
Visible text: Note that the roots of a quadratic equation are the values of that make the equation equal to zero. When we convert the equation to its factored form, we can easily find its roots.
## Basic Principles of Factorization
A quadratic equation in standard form is written as:
```math
ax^2 + bx + c = 0
```
where $$a$$, $$b$$, and $$c$$ are constants and $$a \neq 0$$.
Visible text: where , , and are constants and .
Factorization is based on the following property: If a product equals zero, then at least one of its factors must equal zero.
This means if $$(x - p)(x - q) = 0$$, then:
Visible text: This means if , then:
- $$x - p = 0$$ or $$x - q = 0$$
- Therefore $$x = p$$ or $$x = q$$
Visible text: - or
- Therefore or
## Steps for Factoring Quadratic Equations
Here are the general steps to factor a quadratic equation $$ax^2 + bx + c = 0$$:
Visible text: Here are the general steps to factor a quadratic equation :
1. Ensure the equation is in standard form with the right side equal to zero
2. Find two numbers that when multiplied give $$ac$$ and when added give $$b$$
3. Write the equation in factored form
4. Determine the roots of the equation from these factors
Visible text: 1. Ensure the equation is in standard form with the right side equal to zero
2. Find two numbers that when multiplied give and when added give
3. Write the equation in factored form
4. Determine the roots of the equation from these factors
### Examples of factoring quadratic equations
1. Factoring the equation:
```math
x^2 + 5x + 6 = 0
```
In this equation, $$a = 1$$, $$b = 5$$, and $$c = 6$$.
**Step** $$1$$: The equation is already in standard form with the right side equal to zero.
**Step** $$2$$: We need to find two numbers that:
- When multiplied give $$ac = 1 \times 6 = 6$$
- When added give $$b = 5$$
Factors of $$6$$ are $$1$$, $$2$$, $$3$$, and $$6$$.
Possible factor pairs are $$(1, 6)$$ and $$(2, 3)$$.
The pair $$(2, 3)$$ gives a sum of $$5$$, which matches the value of $$b$$.
**Step** $$3$$: We can write the equation as:
```math
x^2 + 2x + 3x + 6 = 0
```
```math
x(x + 2) + 3(x + 2) = 0
```
```math
(x + 3)(x + 2) = 0
```
**Step** $$4$$: From the factored form above, we get:
- $$x + 3 = 0$$ → $$x = -3$$
- $$x + 2 = 0$$ → $$x = -2$$
Therefore, the roots of the equation are $$x = -3$$ and $$x = -2$$.
2. Factoring the equation:
```math
3x^2 + 13x - 10 = 0
```
In this equation, $$a = 3$$, $$b = 13$$, and $$c = -10$$.
**Step** $$1$$: The equation is already in standard form with the right side equal to zero.
**Step** $$2$$: We need to find two numbers that:
- When multiplied give $$ac = 3 \times (-10) = -30$$
- When added give $$b = 13$$
Factors of $$-30$$ are pairs of numbers with opposite signs:
$$(1, -30), (-1, 30), (2, -15), (-2, 15), (3, -10), (-3, 10), (5, -6), (-5, 6)$$
The pair $$(15, -2)$$ gives a sum of $$13$$, which matches the value of $$b$$.
**Step** $$3$$: We can write the equation as:
```math
3x^2 + 15x - 2x - 10 = 0
```
We can group the terms:
```math
3x^2 + 15x - 2x - 10 = 0
```
```math
3x(x + 5) - 2(x + 5) = 0
```
```math
(3x - 2)(x + 5) = 0
```
**Step** $$4$$: From the factored form above, we get:
- $$3x - 2 = 0$$ → $$x = \frac{2}{3}$$
- $$x + 5 = 0$$ → $$x = -5$$
Therefore, the roots of the equation are $$x = \frac{2}{3}$$ and $$x = -5$$.
3. Factorization When Coefficient $$a \neq 1$$
When the coefficient $$a$$ is not equal to $$1$$, we need some modifications in the factorization steps. There are several approaches:
**Method Using Factors of $$ac$$**
1. Determine the value of $$ac$$
2. Find a pair of factors of $$ac$$ that when added give $$b$$
3. Use this factor pair to split the term $$bx$$ into two terms
4. Factor by grouping
**Example of Factorization**:
```math
2x^2 + 5x - 3 = 0
```
In this equation, $$a = 2$$, $$b = 5$$, and $$c = -3$$.
**Step** $$1$$: Calculate $$ac = 2 \times (-3) = -6$$
**Step** $$2$$: Find a pair of factors of $$-6$$ that when added give $$5$$:
Factors of $$-6$$: $$(1, -6), (-1, 6), (2, -3), (-2, 3)$$
The pair $$(6, -1)$$ gives a sum of $$5$$, which matches the value of $$b$$.
**Step** $$3$$: Split the term $$5x$$ into $$6x - x$$:
```math
2x^2 + 6x - x - 3 = 0
```
**Step** $$4$$: Factor by grouping:
```math
2x^2 + 6x - x - 3 = 0
```
```math
2x(x + 3) - 1(x + 3) = 0
```
```math
(2x - 1)(x + 3) = 0
```
**Step** $$5$$: Determine the roots of the equation:
- $$2x - 1 = 0$$ → $$x = \frac{1}{2}$$
- $$x + 3 = 0$$ → $$x = -3$$
Therefore, the roots of the equation are $$x = \frac{1}{2}$$ and $$x = -3$$.
Visible text: 1. Factoring the equation:
In this equation, , , and .
**Step** : The equation is already in standard form with the right side equal to zero.
**Step** : We need to find two numbers that:
- When multiplied give
- When added give
Factors of are , , , and .
Possible factor pairs are and .
The pair gives a sum of , which matches the value of .
**Step** : We can write the equation as:
**Step** : From the factored form above, we get:
- →
- →
Therefore, the roots of the equation are and .
2. Factoring the equation:
In this equation, , , and .
**Step** : The equation is already in standard form with the right side equal to zero.
**Step** : We need to find two numbers that:
- When multiplied give
- When added give
Factors of are pairs of numbers with opposite signs:
The pair gives a sum of , which matches the value of .
**Step** : We can write the equation as:
We can group the terms:
**Step** : From the factored form above, we get:
- →
- →
Therefore, the roots of the equation are and .
3. Factorization When Coefficient
When the coefficient is not equal to , we need some modifications in the factorization steps. There are several approaches:
**Method Using Factors of **
1. Determine the value of
2. Find a pair of factors of that when added give
3. Use this factor pair to split the term into two terms
4. Factor by grouping
**Example of Factorization**:
In this equation, , , and .
**Step** : Calculate
**Step** : Find a pair of factors of that when added give :
Factors of :
The pair gives a sum of , which matches the value of .
**Step** : Split the term into :
**Step** : Factor by grouping:
**Step** : Determine the roots of the equation:
- →
- →
Therefore, the roots of the equation are and .
## Factoring When One Root Is Known
If we know one of the roots of a quadratic equation, we can use this information to find the complete factorization.
**Example:** One of the roots of the equation $$2x^2 - bx - 6 = 0$$ is $$6$$
Visible text: **Example:** One of the roots of the equation is
If $$x = 6$$ is a root of the equation, then $$(x - 6)$$ is one of its factors.
Visible text: If is a root of the equation, then is one of its factors.
We can substitute $$x = 6$$ into the original equation:
Visible text: We can substitute into the original equation:
Component: MathContainer
Children:
```math
2(6)^2 - b(6) - 6 = 0
```
```math
2 \times 36 - 6b - 6 = 0
```
```math
72 - 6b - 6 = 0
```
```math
66 = 6b
```
```math
b = 11
```
Now we can write the equation as $$2x^2 - 11x - 6 = 0$$.
Visible text: Now we can write the equation as .
Using the factorization method, we factor it as:
Component: MathContainer
Children:
```math
2x^2 - 11x - 6 = 0
```
```math
2x^2 - 12x + x - 6 = 0
```
```math
2x(x - 6) + 1(x - 6) = 0
```
```math
(2x + 1)(x - 6) = 0
```
The roots of the equation are $$x = -\frac{1}{2}$$ and $$x = 6$$.
Visible text: The roots of the equation are and .
## Special Cases of Factorization
1. Form $$ax^2 + bx = 0$$
For equations without a constant term, we can factor out $$x$$ directly:
```math
ax^2 + bx = 0
```
```math
x(ax + b) = 0
```
The roots are $$x = 0$$ and $$x = -\frac{b}{a}$$.
**Example:** $$2x^2 - 18x = 0$$
```math
2x^2 - 18x = 0
```
```math
2x(x - 9) = 0
```
The roots are $$x = 0$$ and $$x = 9$$.
2. Form $$ax^2 - c = 0$$
For equations without an $$x$$ term, we can use the difference of squares pattern:
```math
ax^2 - c = 0
```
```math
ax^2 = c
```
```math
x^2 = \frac{c}{a}
```
```math
x = \pm \sqrt{\frac{c}{a}}
```
**Example:** $$2x^2 - 18 = 0$$
```math
2x^2 - 18 = 0
```
```math
2x^2 = 18
```
```math
x^2 = 9
```
```math
x = \pm 3
```
The roots are $$x = 3$$ and $$x = -3$$.
Visible text: 1. Form
For equations without a constant term, we can factor out directly:
The roots are and .
**Example:**
The roots are and .
2. Form
For equations without an term, we can use the difference of squares pattern:
**Example:**
The roots are and .
## Quadratic Equations That Cannot Be Factored
Not all quadratic equations can be easily factored using rational numbers. In such cases, we can use the quadratic formula:
```math
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
```
A quadratic equation can be factored with rational numbers if the discriminant $$b^2 - 4ac$$ is a perfect square.
Visible text: A quadratic equation can be factored with rational numbers if the discriminant is a perfect square.
## Practice Problems
Factor the following quadratic equations:
1. $$x^2 + 6x + 5 = 0$$
2. $$2x^2 + 5x + 2 = 0$$
3. $$3x^2 - x - 2 = 0$$
4. $$2x^2 - 8x + 6 = 0$$
5. $$x^2 - 9 = 0$$
Visible text: 1.
2.
3.
4.
5.
### Answer Key
1. $$x^2 + 6x + 5 = 0$$
**Step** $$1$$: Identify the coefficients
```math
a = 1, b = 6, c = 5
```
**Step** $$2$$: Find two numbers that when multiplied give $$ac = 5$$ and when added give $$b = 6$$
```math
\text{Factors of } 5: 1 \times 5 = 5 \text{ and} 1 + 5 = 6
```
**Step** $$3$$: Factorization
```math
x^2 + 6x + 5 = 0
```
```math
x^2 + x + 5x + 5 = 0
```
```math
x(x + 1) + 5(x + 1) = 0
```
```math
(x + 5)(x + 1) = 0
```
**Step** $$4$$: Determine the roots of the equation
```math
x + 5 = 0 \Rightarrow x = -5
```
```math
x + 1 = 0 \Rightarrow x = -1
```
Therefore, the roots of the equation are $$x = -5$$ and $$x = -1$$.
2. $$2x^2 + 5x + 2 = 0$$
**Step** $$1$$: Identify the coefficients
```math
a = 2, b = 5, c = 2
```
**Step** $$2$$: Find two numbers that when multiplied give $$ac = 2 \times 2 = 4$$ and when added give $$b = 5$$
```math
\text{Factors of } 4: 1 \times 4 = 4 \text{ and} 1 + 4 = 5
```
**Step** $$3$$: Factorization
```math
2x^2 + 5x + 2 = 0
```
```math
2x^2 + x + 4x + 2 = 0
```
```math
x(2x + 1) + 2(2x + 1) = 0
```
```math
(2x + 1)(x + 2) = 0
```
**Step** $$4$$: Determine the roots of the equation
```math
2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
```
```math
x + 2 = 0 \Rightarrow x = -2
```
Therefore, the roots of the equation are $$x = -\frac{1}{2}$$ and $$x = -2$$.
3. $$3x^2 - x - 2 = 0$$
**Step** $$1$$: Identify the coefficients
```math
a = 3, b = -1, c = -2
```
**Step** $$2$$: Find two numbers that when multiplied give $$ac = 3 \times (-2) = -6$$ and when added give $$b = -1$$
```math
\text{Factors of } -6: (-3) \times 2 = -6 \text{ and} (-3) + 2 = -1
```
**Step** $$3$$: Factorization
```math
3x^2 - x - 2 = 0
```
```math
3x^2 - 3x + 2x - 2 = 0
```
```math
3x(x - 1) + 2(x - 1) = 0
```
```math
(3x + 2)(x - 1) = 0
```
**Step** $$4$$: Determine the roots of the equation
```math
3x + 2 = 0 \Rightarrow x = -\frac{2}{3}
```
```math
x - 1 = 0 \Rightarrow x = 1
```
Therefore, the roots of the equation are $$x = -\frac{2}{3}$$ and $$x = 1$$.
4. $$2x^2 - 8x + 6 = 0$$
**Step** $$1$$: Identify the coefficients
```math
a = 2, b = -8, c = 6
```
**Step** $$2$$: Find two numbers that when multiplied give $$ac = 2 \times 6 = 12$$ and when added give $$b = -8$$
```math
\text{Factors of } 12: (-6) \times (-2) = 12 \text{ and} (-6) + (-2) = -8
```
**Step** $$3$$: Factorization
```math
2x^2 - 8x + 6 = 0
```
```math
2x^2 - 6x - 2x + 6 = 0
```
```math
2x(x - 3) - 2(x - 3) = 0
```
```math
(2x - 2)(x - 3) = 0
```
```math
2(x - 1)(x - 3) = 0
```
**Step** $$4$$: Determine the roots of the equation
```math
x - 1 = 0 \Rightarrow x = 1
```
```math
x - 3 = 0 \Rightarrow x = 3
```
Therefore, the roots of the equation are $$x = 1$$ and $$x = 3$$.
5. $$x^2 - 9 = 0$$
**Step** $$1$$: Identify as a difference of squares
```math
x^2 - 9 = x^2 - 3^2
```
**Step** $$2$$: Use the difference of squares formula $$a^2 - b^2 = (a+b)(a-b)$$
```math
x^2 - 3^2 = (x + 3)(x - 3) = 0
```
**Step** $$3$$: Determine the roots of the equation
```math
x + 3 = 0 \Rightarrow x = -3
```
```math
x - 3 = 0 \Rightarrow x = 3
```
Therefore, the roots of the equation are $$x = -3$$ and $$x = 3$$.
Visible text: 1.
**Step** : Identify the coefficients
**Step** : Find two numbers that when multiplied give and when added give
**Step** : Factorization
**Step** : Determine the roots of the equation
Therefore, the roots of the equation are and .
2.
**Step** : Identify the coefficients
**Step** : Find two numbers that when multiplied give and when added give
**Step** : Factorization
**Step** : Determine the roots of the equation
Therefore, the roots of the equation are and .
3.
**Step** : Identify the coefficients
**Step** : Find two numbers that when multiplied give and when added give
**Step** : Factorization
**Step** : Determine the roots of the equation
Therefore, the roots of the equation are and .
4.
**Step** : Identify the coefficients
**Step** : Find two numbers that when multiplied give and when added give
**Step** : Factorization
**Step** : Determine the roots of the equation
Therefore, the roots of the equation are and .
5.
**Step** : Identify as a difference of squares
**Step** : Use the difference of squares formula
**Step** : Determine the roots of the equation
Therefore, the roots of the equation are and .