Solve 3t-6/11t^2-15/11

$$3 t - \frac { 6 } { 11 } t ^ { 2 } - \frac { 15 } { 11 }$$

Factor

$$\frac{3\left(5-t\right)\left(2t-1\right)}{11}$$

Solution Steps

Factor out \frac{3}{11}.

$$\frac{3\left(11t-2t^{2}-5\right)}{11}$$

Consider 11t-2t^{2}-5. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-2t^{2}+11t-5$$

Factor the expression by grouping. First, the expression needs to be rewritten as -2t^{2}+at+bt-5. To find a and b, set up a system to be solved.

$$a+b=11 ab=-2\left(-5\right)=10$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 11.

$$a=10 b=1$$

Rewrite -2t^{2}+11t-5 as \left(-2t^{2}+10t\right)+\left(t-5\right).

$$\left(-2t^{2}+10t\right)+\left(t-5\right)$$

Factor out 2t in the first and -1 in the second group.

$$2t\left(-t+5\right)-\left(-t+5\right)$$

Factor out common term -t+5 by using distributive property.

$$\left(-t+5\right)\left(2t-1\right)$$

Rewrite the complete factored expression.

$$\frac{3\left(-t+5\right)\left(2t-1\right)}{11}$$

Evaluate

$$-\frac{6t^{2}}{11}+3t-\frac{15}{11}$$