Factorization of Quadratic Trinomials

Quadratic trinomials are expressions of the form ( ax^2 + bx + c ).
To factories, find two numbers that multiply to give ( a \times c ) and add to give ( b ).

Example:
Factorise ( x^2 + 5x + 6 )

Solution:
Numbers that multiply to 6 and add to 5 are 2 and 3.
So,
( x^2 + 5x + 6 = (x + 2)(x + 3) )

Example 1: Factorize ( x^2 + 7x + 10 )

Step 1: Identify the trinomial form
x^2 + 7x + 10

Step 2: Find two numbers whose product is 10 and whose sum is 7.
5 \text{ and } 2

Step 3: Write the factorized form
(x + 5)(x + 2)

Final Answer:
\boxed{(x + 5)(x + 2)}

Example 2: Factorize ( y^2 - 3y - 10 )

Step 1: Identify the trinomial form
y^2 - 3y - 10

Step 2: Find two numbers whose product is −10 and whose sum is −3.
-5 \text{ and } 2

Step 3: Write the factorized form
(y - 5)(y + 2)

Final Answer:
\boxed{(y - 5)(y + 2)}

Example 3: Factorize ( 2x^2 + 7x + 3 )

Step 1: Multiply the coefficient of (x^2) by the constant term
2 \times 3 = 6

Step 2: Find two numbers whose product is 6 and whose sum is 7.
6 \text{ and } 1

Step 3: Rewrite the middle term using 6x and x
2x^2 + 6x + x + 3

Step 4: Group and factor
(2x^2 + 6x) + (x + 3)
2x(x + 3) + 1(x + 3)

Step 5: Factor out the common bracket
(2x + 1)(x + 3)

Final Answer:
\boxed{(2x + 1)(x + 3)}