How can I factorise the equation 6x² + 5x + 6?

Factoring the Quadratic Equation: 6x² + 5x + 6

To factor the quadratic equation 6x² + 5x + 6, we need to find two binomials that multiply to give us this expression. The general form for factoring a quadratic equation is:

ax² + bx + c = (px + q)(rx + s)

Where:

a = 6, b = 5, c = 6

Step 1: Identify the Coefficients

Here, we have:

a = 6
b = 5
c = 6

Step 2: Multiply a and c

Multiply the coefficient of x² (which is 6) with the constant term (which is 6):

6 × 6 = 36

Step 3: Find Two Numbers that Multiply to ac and Add to b

Next, we need to find two numbers that:

Multiply to give 36
Add up to 5

The numbers 9 and -4 satisfy these conditions:

9 × (-4) = -36
9 + (-4) = 5

Step 4: Rewrite the Middle Term

Now, we can rewrite the original quadratic by substituting the middle term with the two numbers found:

6x² + 9x – 4x + 6

Step 5: Factor by Grouping

Next, we can group the terms:

(6x² + 9x) + (-4x + 6)

Factor out the common factors from each group:

3x(2x + 3) – 2(2x + 3)

Step 6: Factor Out the Common Binomial

Now, we can factor out the common binomial factor, which is (2x + 3):

(2x + 3)(3x – 2)

Final Factorization

Thus, the factored form of the quadratic equation 6x² + 5x + 6 is: