Solve p^2-5p+9/4

$$p ^ { 2 } - 5 p + \frac { 9 } { 4 }$$

Factor

$$\frac{\left(2p-9\right)\left(2p-1\right)}{4}$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{4p^{2}-20p+9}{4}$$

Consider 4p^{2}-20p+9. Factor the expression by grouping. First, the expression needs to be rewritten as 4p^{2}+ap+bp+9. To find a and b, set up a system to be solved.

$$a+b=-20 ab=4\times 9=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -20.

$$a=-18 b=-2$$

Rewrite 4p^{2}-20p+9 as \left(4p^{2}-18p\right)+\left(-2p+9\right).

$$\left(4p^{2}-18p\right)+\left(-2p+9\right)$$

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

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

Factor out common term 2p-9 by using distributive property.

$$\left(2p-9\right)\left(2p-1\right)$$

Rewrite the complete factored expression.

$$\frac{\left(2p-9\right)\left(2p-1\right)}{4}$$

Evaluate

$$p^{2}-5p+\frac{9}{4}$$