What is the answer when you factorise 6x² plus 7x plus 12?

Answer 1

The expression:

6x2 + 7x + 12

has no factors. To demonstrate this point, let's try to find two values that we would use to break the expression down into multiple terms. We'll call them a and b. The sum of a and b should be equal to 7 (the coefficient of the middle term), and their product should be 72 (the product of the coefficients of the first and last terms:

a + b = 7

ab = 72

We can then take one of the equations and plug it into the other, reducing to a single variable:

a + 72/a = 7

And then we can try to solve for a:

∴ a - 7 = -72/a

∴ a2 - 7a = -72

∴ a2 - 7a + (7/2)2 = (7/2)2 - 72

∴ (a - 7/2)2 = (7/2)2 - 72

∴ a - 7/2 = (49/4 - 72)1/2

∴ a = (49/4 - 288/4)1/2 + 7/2

∴ a = (49 - 288)1/2 / 2 + 7/2

∴ a = (7 ± i √239) / 2

This leaves us with a pair of complex values for a and b, showing that the original expression can not be factored.