Learn the process to Factorize the Trinomial ax Square Plus bx Plus c. One of the basic expressions for trinomial is ax^{2} + bx + c. To find the ax^{2} + bx + c factors, firstly, we need to find the two numbers and that is p and q. Here, the second term â€˜bâ€™ is the sum of the two numbers that is p + q = b. The product of the first and last terms is equal to the product of two numbers that is p * q = ac. Based on these two instructions, we need to find the values of p and q.

## Steps to Factorize the Trinomial of Form ax^{2} + bx + c?

1. Note down the given expression and compare it with the basic expression ax^{2} + bx + c.

2. Note down the product and sum terms and find the two numbers.

3. Depends on the values of two numbers, expand the given expression.

4. Factor out the common terms.

5. Finally, we will get the product of two terms which is equal to the trinomial expression.

### Solved Examples on Factoring Trinomials of the Form ax^{2} + bx + c

1. Resolve into factors.

(i) 2s^{2} + 9s + 10.

Solution:

The Given expression is 2s^{2} + 9s + 10.

By comparing the given expression 2s^{2} + 9s + 10 with the basic expression ax^{2} + bx + c.

Here, a = 2, b = 9, and c = 10.

The sum of two numbers is p + q = b = 9 = 5 + 4.

The product of two number is p * q = a * c = 2 * 10 = 20 = 5 * 4.

From the above two instructions, we can write the values of two numbers p and q as 5 and 4.

Then, 2s^{2} + 9s + 10 = 2s^{2} + 5s + 4s + 20.

= 2s (s + 5) + 4 (s + 5).

Factor out the common terms.

Then, 2s^{2} + 9s + 10 = (2s + 4) (s + 5).

(ii) 6s^{2} + 7s â€“3

Solution:

The Given expression is 6s^{2} + 7s – 3.

By comparing the given expression 6s^{2} + 7s – 3 with the basic expression ax^{2} + bx + c.

Here, a = 6, b = 7, and c = 3.

The sum of two numbers is p + q = b = 7 = 9 – 2.

The product of two number is p * q = a * c = 6 * 3 = 18 = 9 * 2.

From the above two instructions, we can write the values of two numbers p and q as 9 and 2.

Then, 6s^{2} + 7s -3 = 6s^{2} + 9s – 2s – 3.

= 6s^{2} â€“ 2s + 9s – 3.

= 2s (3s â€“ 1) + 3(3s â€“ 1).

Factor out the common terms.

Then, 6s^{2} + 7s – 3 = (3s – 1) (2s + 3).

2. Factorize the trinomial.

(i) 2x^{2} + 7x + 3.

Solution:

The Given expression is 2x^{2} + 7x + 3.

By comparing the given expression 2x^{2} + 7x + 3 with the basic expression ax^{2} + bx + c.

Here, a = 2, b = 7, and c = 3.

The sum of two numbers is p + q = b = 7 = 6 + 1.

The product of two number is p * q = a * c = 2 * 3 = 6 = 6 * 1.

From the above two instructions, we can write the values of two numbers p and q as 6 and 1.

Then,2x^{2} + 7x + 3 = 2x^{2} + 6x + x + 3.

= 2x (x + 3) + (x + 3).

Factor out the common terms.

Then, 2x^{2} + 7x + 3 = (x + 3) (2x + 1).

(ii) 3s^{2} â€“ 4s â€“ 4.

Solution:

The Given expression is 3s^{2} â€“ 4s â€“ 4.

By comparing the given expression 3s^{2} â€“ 4s â€“ 4 with the basic expression ax^{2} + bx + c.

Here, a = 3, b = – 4, and c = – 4.

The sum of two numbers is p + q = b = – 4 = – 6 + 2.

The product of two number is p * q = a * c = 3 * (- 4) = – 12 = (- 6) * 2.

From the above two instructions, we can write the values of two numbers p and q as – 6 and 2.

Then, 3s^{2} â€“ 4s â€“ 4 = 3s^{2} â€“ 6s + 2s – 4.

= 3s (s â€“ 2) + 2(s â€“ 2).

Factor out the common terms.

Then, 3s^{2} â€“ 4s â€“ 4 = (s – 2) (3s + 2).