Solve 12x^2-4x-3x+1

$$12 x ^ { 2 } - 4 x - 3 x + 1$$

Factor

$$\left(3x-1\right)\left(4x-1\right)$$

Solution Steps

Multiply and combine like terms.

$$12x^{2}-7x+1$$

Factor the expression by grouping. First, the expression needs to be rewritten as 12x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=-7 ab=12\times 1=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-4 b=-3$$

Rewrite 12x^{2}-7x+1 as \left(12x^{2}-4x\right)+\left(-3x+1\right).

$$\left(12x^{2}-4x\right)+\left(-3x+1\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(3x-1\right)-\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(4x-1\right)$$

Evaluate

$$\left(3x-1\right)\left(4x-1\right)$$

Solution Steps

Combine -4x and -3x to get -7x.

$$12x^{2}-7x+1$$