Solve 12t^2-8t-3t+2

$$12 t ^ { 2 } - 8 t - 3 t + 2$$

Factor

$$\left(3t-2\right)\left(4t-1\right)$$

Solution Steps

Multiply and combine like terms.

$$12t^{2}-11t+2$$

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

$$a+b=-11 ab=12\times 2=24$$

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 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-8 b=-3$$

Rewrite 12t^{2}-11t+2 as \left(12t^{2}-8t\right)+\left(-3t+2\right).

$$\left(12t^{2}-8t\right)+\left(-3t+2\right)$$

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

$$4t\left(3t-2\right)-\left(3t-2\right)$$

Factor out common term 3t-2 by using distributive property.

$$\left(3t-2\right)\left(4t-1\right)$$

Evaluate

$$\left(3t-2\right)\left(4t-1\right)$$

Solution Steps

Combine -8t and -3t to get -11t.

$$12t^{2}-11t+2$$