Factorising 6x^2 + 5x - 6 using the factor theorem

To factorise the given expression, we can use the factor theorem. The factor theorem states that if a polynomial f(x) has a factor (x - a), then f(a) will be equal to zero.

Step 1: Identifying potential factors
To factorise the expression 6x^2 + 5x - 6, we need to find two binomial factors that multiply together to give the original expression. Let's consider all possible combinations of factors for the constant term (-6) and the coefficient of x (5).

The factors of -6 are:
-1, 1, -2, 2, -3, 3, -6, 6

Now we need to find two factors that add up to the coefficient of x (5). After examining the combinations, we find that the factors 6 and -1 satisfy this condition since 6 + (-1) = 5.

Step 2: Writing the potential factors
The potential factors are (x + 6) and (x - 1) because they multiply together to give the original expression.

Step 3: Applying the factor theorem
Now, we can use the factor theorem to verify if these potential factors are indeed factors of the expression 6x^2 + 5x - 6. We substitute the value of x from each factor and check if the resulting expression equals zero.

For the factor (x + 6):
f(-6) = 6(-6)^2 + 5(-6) - 6
= 6(36) - 30 - 6
= 216 - 30 - 6
= 180 - 6
= 174

For the factor (x - 1):
f(1) = 6(1)^2 + 5(1) - 6
= 6(1) + 5(1) - 6
= 6 + 5 - 6
= 11 - 6
= 5

Since neither of the potential factors results in zero when substituted, we can conclude that (x + 6) and (x - 1) are not factors of the expression 6x^2 + 5x - 6.

Step 4: Finding the correct factors
Since the potential factors we identified are not factors of the expression, we need to find the correct factors.

Step 5: Factoring the expression
To factorise the expression, we can use the quadratic formula or complete the square method. Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the expression 6x^2 + 5x - 6, the coefficients are:
a = 6
b = 5
c = -6

Substituting these values into the quadratic formula, we have:

x = (-5 ± √(5^2 - 4(6)(-6))) / (2(6))
x = (-5 ± √(25 + 144)) / 12
x = (-5 ± √169)

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