# Factoring Quadratic Equations - Trial and Error

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More Lessons for Algebra, Math Worksheets

There are several techniques that can be used to factor quadratic equations.
In this lesson, we will learn how to factor quadratic equations, where the coefficient of x2 is 1, using the trial and error method. In this method, we will need to try out different possibilities to get the right factors for the given quadratic equation.

We also have another technique that does not need require guessing or trial and error.

### If the Coefficient Of x2 Is 1

To factorize quadratic equations of the form: x2 + bx + c, you will need to find two numbers whose product is c and whose sum is b.

Example 1: (b and c are both positive)

Solve the quadratic equation: x2 + 7x + 10 = 0

Step 1: List out the factors of 10:

1 × 10, 2 × 5

Step 2: Find the factors whose sum is 7:

1 + 10 ≠ 7
2 + 5 = 7

Step 3: Write out the factors and check using the distributive property.

(x + 2) (x + 5) = x2 + 5x + 2x + 10 = x2 + 7x + 10
The factors are (x + 2) (x + 5)

Step 4: Going back to the original quadratic equation

x2 + 7x + 10 = 0       Factorize the left side of the quadratic equation
(x + 2) (x + 5) = 0

We get two values for x.

Answer:   x = – 2, x = – 5

Example 2: (b is positive and c is negative)

Get the values of x for the equation: x2 + 4x – 5 = 0

Step 1: List out the factors of – 5:

1 × –5, –1 × 5

Step 2: Find the factors whose sum is 4:

1 – 5 ≠ 4
–1 + 5 = 4

Step 3: Write out the factors and check using the distributive property.

(x – 1)(x + 5)= x2 + 5xx – 5 = x2 + 4x – 5

Step 4: Going back to the original quadratic equation

x2 + 4x – 5 = 0       Factorize the left hand side of the equation
(x – 1)(x + 5) = 0

We get two values for x.

Answer:   x = 1, x = – 5

Example 3: (b and c are both negative)

Get the values of x for the equation: x2 – 5x – 6

Step 1: List out the factors of – 6:

1 × –6, –1 × 6, 2 × –3, –2 × 3

Step 2: Find the factors whose sum is –5:

1 + ( –6) = –5

Step 3: Write out the factors and check using the distributive property.

(x + 1) (x – 6) = x2 – 6 x + x – 6 = x2 – 5x – 6

Step 4: Going back to the original quadratic equation

x2 – 5x – 6 = 0       Factorize the left hand side of the equation
(x + 1) (x – 6) = 0

We get two values for x.

Answer: x = –1, x = 6

Example 4: (b is negative and c is positive)

Get the values of x for the equation: x2 – 6x + 8 = 0

Step 1: List out the factors of 8:

We need to get the negative factors of 8 to get a negative sum.
–1 × – 8, –2 × –4

Step 2: Find the factors whose sum is – 6:

–1 + ( –8) ≠ –6
–2 + ( –4) = –6

Step 3: Write out the factors and check using the distributive property.

(x – 2) (x – 4) = x2 – 4 x – 2x + 8 = x2 – 6x + 8

Step 4: Going back to the original quadratic equation

x2 – 6x + 8 = 0       Factorize the left hand side of the equation
(x – 2) (x – 4) = 0

We get two values for x.

Answer: x = 2, x = 4

Examples of solving quadratic equations
Examples:
1. Solve x2 + 6x + 8 = 0
2. Solve 2x2 + 20x + 50 = 0
3. Solve x2 - x - 30 = 0
Factoring Quadratic Expressions
coefficient of x2 = −1
Examples:
1. Factor x2 + 10x + 9
2. Factor x2 - 11x + 24
3. Factor x2 - x - 56
3. Factor -x2 - 5x + 24
3. Factor -x2 + 18x - 72
Solve quadratic equation by factoring
Example:
Solve x2 - 3x = 4
How to factor quadratics?
Examples:
1) x2 - 3x - 18
2) x2 - 6x - 16
Related Topics:
Factoring Out Common Factors (GCF).
Factoring Quadratic Equations using Perfect Square Trinomial (Square of a Sum or Square of a Difference) or Difference of Two Squares.
Factoring Quadratic Equations where the coefficient of x2 is greater than 1
Factoring Quadratic Equations by Completing the Square
Factoring Quadratic Equations using the Quadratic Formula.

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