Solve x^2-135x+162*27

$${ x }^{ 2 } -135x+162 \times 27$$

Factor

$$\left(x-81\right)\left(x-54\right)$$

Solution Steps

Multiply and combine like terms.

$$x^{2}-135x+4374$$

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+4374. To find a and b, set up a system to be solved.

$$a+b=-135 ab=1\times 4374=4374$$

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 4374.

$$-1,-4374 -2,-2187 -3,-1458 -6,-729 -9,-486 -18,-243 -27,-162 -54,-81$$

Calculate the sum for each pair.

$$-1-4374=-4375 -2-2187=-2189 -3-1458=-1461 -6-729=-735 -9-486=-495 -18-243=-261 -27-162=-189 -54-81=-135$$

The solution is the pair that gives sum -135.

$$a=-81 b=-54$$

Rewrite x^{2}-135x+4374 as \left(x^{2}-81x\right)+\left(-54x+4374\right).

$$\left(x^{2}-81x\right)+\left(-54x+4374\right)$$

Factor out x in the first and -54 in the second group.

$$x\left(x-81\right)-54\left(x-81\right)$$

Factor out common term x-81 by using distributive property.

$$\left(x-81\right)\left(x-54\right)$$

Evaluate

$$\left(x-81\right)\left(x-54\right)$$

Solution Steps

Multiply 162 and 27 to get 4374.

$$x^{2}-135x+4374$$