Solve x^2-1/12x+7x+x^2=1/7

$$\frac{ { x }^{ 2 } -1 }{ 12x+7x+ { x }^{ 2 } } = \frac{ 1 }{ 7 }$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -19,0 since division by zero is not defined. Multiply both sides of the equation by 7x\left(x+19\right), the least common multiple of 12x+7x+x^{2},7.

$$7\left(x^{2}-1\right)=x\left(x+19\right)$$

Use the distributive property to multiply 7 by x^{2}-1.

$$7x^{2}-7=x\left(x+19\right)$$

Use the distributive property to multiply x by x+19.

$$7x^{2}-7=x^{2}+19x$$

Subtract x^{2} from both sides.

$$7x^{2}-7-x^{2}=19x$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-7=19x$$

Subtract 19x from both sides.

$$6x^{2}-7-19x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$6x^{2}-19x-7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=-19 ab=6\left(-7\right)=-42$$

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 -42.

$$1,-42 2,-21 3,-14 6,-7$$

Calculate the sum for each pair.

$$1-42=-41 2-21=-19 3-14=-11 6-7=-1$$

The solution is the pair that gives sum -19.

$$a=-21 b=2$$

Rewrite 6x^{2}-19x-7 as \left(6x^{2}-21x\right)+\left(2x-7\right).

$$\left(6x^{2}-21x\right)+\left(2x-7\right)$$

Factor out 3x in 6x^{2}-21x.

$$3x\left(2x-7\right)+2x-7$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(3x+1\right)$$

To find equation solutions, solve 2x-7=0 and 3x+1=0.

$$x=\frac{7}{2} x=-\frac{1}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -19,0 since division by zero is not defined. Multiply both sides of the equation by 7x\left(x+19\right), the least common multiple of 12x+7x+x^{2},7.

$$7\left(x^{2}-1\right)=x\left(x+19\right)$$

Use the distributive property to multiply 7 by x^{2}-1.

$$7x^{2}-7=x\left(x+19\right)$$

Use the distributive property to multiply x by x+19.

$$7x^{2}-7=x^{2}+19x$$

Subtract x^{2} from both sides.

$$7x^{2}-7-x^{2}=19x$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-7=19x$$

Subtract 19x from both sides.

$$6x^{2}-7-19x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}-19x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -19 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 6\left(-7\right)}}{2\times 6}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 6\left(-7\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-19\right)±\sqrt{361-24\left(-7\right)}}{2\times 6}$$

Multiply -24 times -7.

$$x=\frac{-\left(-19\right)±\sqrt{361+168}}{2\times 6}$$

Add 361 to 168.

$$x=\frac{-\left(-19\right)±\sqrt{529}}{2\times 6}$$

Take the square root of 529.

$$x=\frac{-\left(-19\right)±23}{2\times 6}$$

The opposite of -19 is 19.

$$x=\frac{19±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{19±23}{12}$$

Now solve the equation x=\frac{19±23}{12} when ± is plus. Add 19 to 23.

$$x=\frac{42}{12}$$

Reduce the fraction \frac{42}{12} to lowest terms by extracting and canceling out 6.

$$x=\frac{7}{2}$$

Now solve the equation x=\frac{19±23}{12} when ± is minus. Subtract 23 from 19.

$$x=-\frac{4}{12}$$

Reduce the fraction \frac{-4}{12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{3}$$

The equation is now solved.

$$x=\frac{7}{2} x=-\frac{1}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -19,0 since division by zero is not defined. Multiply both sides of the equation by 7x\left(x+19\right), the least common multiple of 12x+7x+x^{2},7.

$$7\left(x^{2}-1\right)=x\left(x+19\right)$$

Use the distributive property to multiply 7 by x^{2}-1.

$$7x^{2}-7=x\left(x+19\right)$$

Use the distributive property to multiply x by x+19.

$$7x^{2}-7=x^{2}+19x$$

Subtract x^{2} from both sides.

$$7x^{2}-7-x^{2}=19x$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-7=19x$$

Subtract 19x from both sides.

$$6x^{2}-7-19x=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$6x^{2}-19x=7$$

Divide both sides by 6.

$$\frac{6x^{2}-19x}{6}=\frac{7}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{19}{6}x=\frac{7}{6}$$

Divide -\frac{19}{6}, the coefficient of the x term, by 2 to get -\frac{19}{12}. Then add the square of -\frac{19}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{6}x+\left(-\frac{19}{12}\right)^{2}=\frac{7}{6}+\left(-\frac{19}{12}\right)^{2}$$

Square -\frac{19}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{6}x+\frac{361}{144}=\frac{7}{6}+\frac{361}{144}$$

Add \frac{7}{6} to \frac{361}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{6}x+\frac{361}{144}=\frac{529}{144}$$

Factor x^{2}-\frac{19}{6}x+\frac{361}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x-\frac{19}{12}=\frac{23}{12} x-\frac{19}{12}=-\frac{23}{12}$$

Add \frac{19}{12} to both sides of the equation.

$$x=\frac{7}{2} x=-\frac{1}{3}$$