Solve x^2-x/5-12/25

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

Factor

$$\frac{\left(5x-4\right)\left(5x+3\right)}{25}$$

Solution Steps

Factor out \frac{1}{25}.

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

Consider 25x^{2}-5x-12. Factor the expression by grouping. First, the expression needs to be rewritten as 25x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-5 ab=25\left(-12\right)=-300$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -300.

$$1,-300 2,-150 3,-100 4,-75 5,-60 6,-50 10,-30 12,-25 15,-20$$

Calculate the sum for each pair.

$$1-300=-299 2-150=-148 3-100=-97 4-75=-71 5-60=-55 6-50=-44 10-30=-20 12-25=-13 15-20=-5$$

The solution is the pair that gives sum -5.

$$a=-20 b=15$$

Rewrite 25x^{2}-5x-12 as \left(25x^{2}-20x\right)+\left(15x-12\right).

$$\left(25x^{2}-20x\right)+\left(15x-12\right)$$

Factor out 5x in the first and 3 in the second group.

$$5x\left(5x-4\right)+3\left(5x-4\right)$$

Factor out common term 5x-4 by using distributive property.

$$\left(5x-4\right)\left(5x+3\right)$$

Rewrite the complete factored expression.

$$\frac{\left(5x-4\right)\left(5x+3\right)}{25}$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 25 is 25. Multiply \frac{x}{5} times \frac{5}{5}.

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

Since -\frac{5x}{25} and \frac{12}{25} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{25}{25}.

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

Since \frac{25x^{2}}{25} and \frac{-5x-12}{25} have the same denominator, add them by adding their numerators.

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