Solve =x^2-11x+121/4-121/4+18

$$= x ^ { 2 } - 11 x + \frac { 121 } { 4 } - \frac { 121 } { 4 } + 18$$

Factor

$$\left(x-9\right)\left(x-2\right)$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{4x^{2}-44x+72}{4}$$

Consider 4x^{2}-44x+121-121+72. Multiply and combine like terms.

$$4x^{2}-44x+72$$

Consider 4x^{2}-44x+72. Factor out 4.

$$4\left(x^{2}-11x+18\right)$$

Consider x^{2}-11x+18. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=-11 ab=1\times 18=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -11.

$$a=-9 b=-2$$

Rewrite x^{2}-11x+18 as \left(x^{2}-9x\right)+\left(-2x+18\right).

$$\left(x^{2}-9x\right)+\left(-2x+18\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(x-9\right)-2\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(x-2\right)$$

Evaluate

$$\left(x-9\right)\left(x-2\right)$$

Solution Steps

Subtract \frac{121}{4} from \frac{121}{4} to get 0.

$$x^{2}-11x+18$$