Question

Find the roots of 4x2-5x-1

Answer 1

Answer:

Here is a method that I always teach my students.

Given ax2 + bx + c = 0, find two integers with b as the sum and ac as the product.

With 4x2 - 5x + 1 = 0, for example, you want two integers with a sum of -5 and a product of 4*1 = 4.

The only pairs of integers (order doesn't matter) with a product 4 are

(1,4), (2,2), (-1,-4), (-2,-2)

The only pair in this list with a sum of -5 is (-1,-4)

You can now factor by grouping:

4x2 - 5x + 1 = 4x2 - 4x - x + 1 = 4x(x - 1) - 1(x - 1) = (4x - 1)(x - 1).

(4x - 1)(x - 1) = 0

4x - 1 = 0 or x - 1 = 0

x = 1/4 or x = 1

Answer 2

Answer:

Given ax2 + bx + c = 0, find two integers with b as the sum and ac as the product. With 4x2 - 5x + 1 = 0, for example, you want two integers with a sum of -5 and a product of 4*1 = 4. You can now factor by grouping: 4x2 - 5x + 1 = 4x2 - 4x - x + 1 = 4x(x - 1) - 1(x - 1) = (4x - 1)(x - 1).