Solve 7x^2/7-6x/7-1/7

$$\frac{ 7 { x }^{ 2 } }{ 7 } - \frac{ 6x }{ 7 } - \frac{ 1 }{ 7 }$$

Evaluate

$$x^{2}-\frac{6x}{7}-\frac{1}{7}$$

Short Solution Steps

Cancel out 7 and 7.

$$x^{2}-\frac{6x}{7}-\frac{1}{7}$$

Since -\frac{6x}{7} and \frac{1}{7} have the same denominator, subtract them by subtracting their numerators.

$$x^{2}+\frac{-6x-1}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{7}{7}.

$$\frac{7x^{2}}{7}+\frac{-6x-1}{7}$$

Since \frac{7x^{2}}{7} and \frac{-6x-1}{7} have the same denominator, add them by adding their numerators.

$$\frac{7x^{2}-6x-1}{7}$$

Factor

$$\frac{\left(x-1\right)\left(7x+1\right)}{7}$$

Solution Steps

Factor out \frac{1}{7}.

$$\frac{7x^{2}-6x-1}{7}$$

Consider 7x^{2}-6x-1. Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-6 ab=7\left(-1\right)=-7$$

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. The only such pair is the system solution.

$$a=-7 b=1$$

Rewrite 7x^{2}-6x-1 as \left(7x^{2}-7x\right)+\left(x-1\right).

$$\left(7x^{2}-7x\right)+\left(x-1\right)$$

Factor out 7x in 7x^{2}-7x.

$$7x\left(x-1\right)+x-1$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(7x+1\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x-1\right)\left(7x+1\right)}{7}$$