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Solve by factoring. 4x2 - 5x + 1 = 0

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2 Answers

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

I find it easiest to start by looking at the third term. When factoring, the third term is the product of two integers (no variable). Because it is 1, you know that the only terms it could factor into are 1. Now looking at the (+/-) values in the equation. Since the middle term is negative and the last term is positive, this lets you know that the two integers that are multiplied together are negative. So you get:

(4x-1)(x-1)