Solve 1/5x^2-12/25x+4/25

$$\frac{ 1 }{ 5 } { x }^{ 2 } - \frac{ 12 }{ 25 } x+ \frac{ 4 }{ 25 }$$

Factor

$$\frac{\left(x-2\right)\left(5x-2\right)}{25}$$

Solution Steps

Factor out \frac{1}{25}.

$$\frac{5x^{2}-12x+4}{25}$$

Consider 5x^{2}-12x+4. Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-12 ab=5\times 4=20$$

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 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -12.

$$a=-10 b=-2$$

Rewrite 5x^{2}-12x+4 as \left(5x^{2}-10x\right)+\left(-2x+4\right).

$$\left(5x^{2}-10x\right)+\left(-2x+4\right)$$

Factor out 5x in the first and -2 in the second group.

$$5x\left(x-2\right)-2\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(5x-2\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x-2\right)\left(5x-2\right)}{25}$$

Evaluate

$$\frac{\left(x-2\right)\left(5x-2\right)}{25}$$