Solve 5s^2-9s+45s-81

$$5 s ^ { 2 } - 9 s + 45 s - 81$$

Factor

$$\left(5s-9\right)\left(s+9\right)$$

Solution Steps

Multiply and combine like terms.

$$5s^{2}+36s-81$$

Factor the expression by grouping. First, the expression needs to be rewritten as 5s^{2}+as+bs-81. To find a and b, set up a system to be solved.

$$a+b=36 ab=5\left(-81\right)=-405$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -405.

$$-1,405 -3,135 -5,81 -9,45 -15,27$$

Calculate the sum for each pair.

$$-1+405=404 -3+135=132 -5+81=76 -9+45=36 -15+27=12$$

The solution is the pair that gives sum 36.

$$a=-9 b=45$$

Rewrite 5s^{2}+36s-81 as \left(5s^{2}-9s\right)+\left(45s-81\right).

$$\left(5s^{2}-9s\right)+\left(45s-81\right)$$

Factor out s in the first and 9 in the second group.

$$s\left(5s-9\right)+9\left(5s-9\right)$$

Factor out common term 5s-9 by using distributive property.

$$\left(5s-9\right)\left(s+9\right)$$

Evaluate

$$\left(5s-9\right)\left(s+9\right)$$

Solution Steps

Combine -9s and 45s to get 36s.

$$5s^{2}+36s-81$$