Quadratic Equation Factorization

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.

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.

Basic Principles of Factorization

A quadratic equation in standard form is written as:

where $a$, $b$, and $c$ are constants and $a \neq 0$.

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:

• $x - p = 0$ or $x - q = 0$
• Therefore $x = p$ or $x = q$

Steps for Factoring Quadratic Equations

Here are the general steps to factor a quadratic equation $ax^2 + bx + c = 0$:

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

Examples of factoring quadratic equations

1. Factoring the equation:

   $$x^2 + 5x + 6 = 0$$

   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: (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:

   $$x^2 + 2x + 3x + 6 = 0$$

   $$x(x + 2) + 3(x + 2) = 0$$

   $$(x + 3)(x + 2) = 0$$

   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:

   $$3x^2 + 13x - 10 = 0$$

   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:

   $$3x^2 + 15x - 2x - 10 = 0$$

   We can group the terms:

   $$3x^2 + 15x - 2x - 10 = 0$$

   $$3x(x + 5) - 2(x + 5) = 0$$

   $$(3x - 2)(x + 5) = 0$$

   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:

   $$2x^2 + 5x - 3 = 0$$

   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 $6x - x$:

   $$2x^2 + 6x - x - 3 = 0$$

   Step 4: Factor by grouping:

   $$2x^2 + 6x - x - 3 = 0$$

   $$2x(x + 3) - 1(x + 3) = 0$$

   $$(2x - 1)(x + 3) = 0$$

   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 $x = -3$.

Quick Method: When We Know One of the Roots

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

If $x = 6$ is a root of the equation, then $(x - 6)$ is one of its factors.

We can substitute $x = 6$ into the original equation:

Now we can write the equation as $2x^2 - 11x - 6 = 0$.

Using the factorization method, we factor it as:

The roots of the equation are $x = -\frac{1}{2}$ and $x = 6$.

Special Cases of Factorization

1. Form $ax^2 + bx = 0$

   For equations without a constant term, we can factor out $x$ directly:

   $$ax^2 + bx = 0$$

   $$x(ax + b) = 0$$

   The roots are $x = 0$ and $x = -\frac{b}{a}$.

   Example: $2x^2 - 18x = 0$

   $$2x^2 - 18x = 0$$

   $$2x(x - 9) = 0$$

   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:

   $$ax^2 - c = 0$$

   $$ax^2 = c$$

   $$x^2 = \frac{c}{a}$$

   $$x = \pm \sqrt{\frac{c}{a}}$$

   Example: $2x^2 - 18 = 0$

   $$2x^2 - 18 = 0$$

   $$2x^2 = 18$$

   $$x^2 = 9$$

   $$x = \pm 3$$

   The roots are $x = 3$ and $x = -3$.

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:

A quadratic equation can be factored with rational numbers if the discriminant $b^2 - 4ac$ 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. $x^2 - 9 = 0$

Answer Key

1. $x^2 + 6x + 5 = 0$

   Step 1: Identify the coefficients

   $$a = 1, b = 6, c = 5$$

   Step 2: Find two numbers that when multiplied give $ac = 5$ and when added give $b = 6$

   $$\text{Factors of } 5: 1 \times 5 = 5 \text{ and } 1 + 5 = 6$$

   Step 3: Factorization

   $$x^2 + 6x + 5 = 0$$

   $$x^2 + x + 5x + 5 = 0$$

   $$x(x + 1) + 5(x + 1) = 0$$

   $$(x + 5)(x + 1) = 0$$

   Step 4: Determine the roots of the equation

   $$x + 5 = 0 \Rightarrow x = -5$$

   $$x + 1 = 0 \Rightarrow x = -1$$

   Therefore, the roots of the equation are $x = -5$ and $x = -1$.

2. $2x^2 + 5x + 2 = 0$

   Step 1: Identify the coefficients

   $$a = 2, b = 5, c = 2$$

   Step 2: Find two numbers that when multiplied give $ac = 2 \times 2 = 4$ and when added give $b = 5$

   $$\text{Factors of } 4: 1 \times 4 = 4 \text{ and } 1 + 4 = 5$$

   Step 3: Factorization

   $$2x^2 + 5x + 2 = 0$$

   $$2x^2 + x + 4x + 2 = 0$$

   $$x(2x + 1) + 2(2x + 1) = 0$$

   $$(2x + 1)(x + 2) = 0$$

   Step 4: Determine the roots of the equation

   $$2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$$

   $$x + 2 = 0 \Rightarrow x = -2$$

   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

   $$a = 3, b = -1, c = -2$$

   Step 2: Find two numbers that when multiplied give $ac = 3 \times (-2) = -6$ and when added give $b = -1$

   $$\text{Factors of } -6: (-3) \times 2 = -6 \text{ and } (-3) + 2 = -1$$

   Step 3: Factorization

   $$3x^2 - x - 2 = 0$$

   $$3x^2 - 3x + 2x - 2 = 0$$

   $$3x(x - 1) + 2(x - 1) = 0$$

   $$(3x + 2)(x - 1) = 0$$

   Step 4: Determine the roots of the equation

   $$3x + 2 = 0 \Rightarrow x = -\frac{2}{3}$$

   $$x - 1 = 0 \Rightarrow x = 1$$

   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

   $$a = 2, b = -8, c = 6$$

   Step 2: Find two numbers that when multiplied give $ac = 2 \times 6 = 12$ and when added give $b = -8$

   $$\text{Factors of } 12: (-6) \times (-2) = 12 \text{ and } (-6) + (-2) = -8$$

   Step 3: Factorization

   $$2x^2 - 8x + 6 = 0$$

   $$2x^2 - 6x - 2x + 6 = 0$$

   $$2x(x - 3) - 2(x - 3) = 0$$

   $$(2x - 2)(x - 3) = 0$$

   $$2(x - 1)(x - 3) = 0$$

   Step 4: Determine the roots of the equation

   $$x - 1 = 0 \Rightarrow x = 1$$

   $$x - 3 = 0 \Rightarrow x = 3$$

   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

   $$x^2 - 9 = x^2 - 3^2$$

   Step 2: Use the difference of squares formula $a^2 - b^2 = (a+b)(a-b)$

   $$x^2 - 3^2 = (x + 3)(x - 3) = 0$$

   Step 3: Determine the roots of the equation

   $$x + 3 = 0 \Rightarrow x = -3$$

   $$x - 3 = 0 \Rightarrow x = 3$$

   Therefore, the roots of the equation are $x = -3$ and $x = 3$.