Solve 7x/x-7x=1/7x

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

Solve for x

$$x=\frac{3\sqrt{5}}{14}+\frac{1}{2}\approx 0.979157424$$

$$x=-\frac{3\sqrt{5}}{14}+\frac{1}{2}\approx 0.020842576$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x, the least common multiple of x,7x.

$$7\times 7x-7x\times 7x=1$$

Multiply 7 and 7 to get 49.

$$49x-7x\times 7x=1$$

Multiply x and x to get x^{2}.

$$49x-7x^{2}\times 7=1$$

Multiply -7 and 7 to get -49.

$$49x-49x^{2}=1$$

Subtract 1 from both sides.

$$49x-49x^{2}-1=0$$

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.

$$-49x^{2}+49x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, 49 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-49±\sqrt{49^{2}-4\left(-49\right)\left(-1\right)}}{2\left(-49\right)}$$

Square 49.

$$x=\frac{-49±\sqrt{2401-4\left(-49\right)\left(-1\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-49±\sqrt{2401+196\left(-1\right)}}{2\left(-49\right)}$$

Multiply 196 times -1.

$$x=\frac{-49±\sqrt{2401-196}}{2\left(-49\right)}$$

Add 2401 to -196.

$$x=\frac{-49±\sqrt{2205}}{2\left(-49\right)}$$

Take the square root of 2205.

$$x=\frac{-49±21\sqrt{5}}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{-49±21\sqrt{5}}{-98}$$

Now solve the equation x=\frac{-49±21\sqrt{5}}{-98} when ± is plus. Add -49 to 21\sqrt{5}.

$$x=\frac{21\sqrt{5}-49}{-98}$$

Divide -49+21\sqrt{5} by -98.

$$x=-\frac{3\sqrt{5}}{14}+\frac{1}{2}$$

Now solve the equation x=\frac{-49±21\sqrt{5}}{-98} when ± is minus. Subtract 21\sqrt{5} from -49.

$$x=\frac{-21\sqrt{5}-49}{-98}$$

Divide -49-21\sqrt{5} by -98.

$$x=\frac{3\sqrt{5}}{14}+\frac{1}{2}$$

The equation is now solved.

$$x=-\frac{3\sqrt{5}}{14}+\frac{1}{2} x=\frac{3\sqrt{5}}{14}+\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x, the least common multiple of x,7x.

$$7\times 7x-7x\times 7x=1$$

Multiply 7 and 7 to get 49.

$$49x-7x\times 7x=1$$

Multiply x and x to get x^{2}.

$$49x-7x^{2}\times 7=1$$

Multiply -7 and 7 to get -49.

$$49x-49x^{2}=1$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-49x^{2}+49x=1$$

Divide both sides by -49.

$$\frac{-49x^{2}+49x}{-49}=\frac{1}{-49}$$

Dividing by -49 undoes the multiplication by -49.

$$x^{2}+\frac{49}{-49}x=\frac{1}{-49}$$

Divide 49 by -49.

$$x^{2}-x=\frac{1}{-49}$$

Divide 1 by -49.

$$x^{2}-x=-\frac{1}{49}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{1}{49}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=-\frac{1}{49}+\frac{1}{4}$$

Add -\frac{1}{49} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{45}{196}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{45}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{45}{196}}$$

Simplify.

$$x-\frac{1}{2}=\frac{3\sqrt{5}}{14} x-\frac{1}{2}=-\frac{3\sqrt{5}}{14}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{3\sqrt{5}}{14}+\frac{1}{2} x=-\frac{3\sqrt{5}}{14}+\frac{1}{2}$$