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) = 0We 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 + 5x – x – 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) = 0We 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) = 0We 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) = 0We get two values for x.
Answer: x = 2, x = 4
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