Solve 2/3(-x+frac{5{18})}+1/3x=7

$$\frac{ 2 }{ 3(-x+ \frac{ 5 }{ 18 } ) } + \frac{ 1 }{ 3x } =7$$

Solve for x

$$x=\frac{1}{9}\approx 0.111111111$$

$$x=\frac{5}{42}\approx 0.119047619$$

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 3x.

$$18x\left(-18x+5\right)^{-1}\times 2+1=21x$$

Multiply 18 and 2 to get 36.

$$36x\left(-18x+5\right)^{-1}+1=21x$$

Subtract 21x from both sides.

$$36x\left(-18x+5\right)^{-1}+1-21x=0$$

Reorder the terms.

$$-21x+36\times \frac{1}{-18x+5}x+1=0$$

Variable x cannot be equal to \frac{5}{18} since division by zero is not defined. Multiply both sides of the equation by -18x+5.

$$-21x\left(-18x+5\right)+36\times 1x-18x+5=0$$

Use the distributive property to multiply -21x by -18x+5.

$$378x^{2}-105x+36\times 1x-18x+5=0$$

Multiply 36 and 1 to get 36.

$$378x^{2}-105x+36x-18x+5=0$$

Combine -105x and 36x to get -69x.

$$378x^{2}-69x-18x+5=0$$

Combine -69x and -18x to get -87x.

$$378x^{2}-87x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 378 for a, -87 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-87\right)±\sqrt{\left(-87\right)^{2}-4\times 378\times 5}}{2\times 378}$$

Square -87.

$$x=\frac{-\left(-87\right)±\sqrt{7569-4\times 378\times 5}}{2\times 378}$$

Multiply -4 times 378.

$$x=\frac{-\left(-87\right)±\sqrt{7569-1512\times 5}}{2\times 378}$$

Multiply -1512 times 5.

$$x=\frac{-\left(-87\right)±\sqrt{7569-7560}}{2\times 378}$$

Add 7569 to -7560.

$$x=\frac{-\left(-87\right)±\sqrt{9}}{2\times 378}$$

Take the square root of 9.

$$x=\frac{-\left(-87\right)±3}{2\times 378}$$

The opposite of -87 is 87.

$$x=\frac{87±3}{2\times 378}$$

Multiply 2 times 378.

$$x=\frac{87±3}{756}$$

Now solve the equation x=\frac{87±3}{756} when ± is plus. Add 87 to 3.

$$x=\frac{90}{756}$$

Reduce the fraction \frac{90}{756} to lowest terms by extracting and canceling out 18.

$$x=\frac{5}{42}$$

Now solve the equation x=\frac{87±3}{756} when ± is minus. Subtract 3 from 87.

$$x=\frac{84}{756}$$

Reduce the fraction \frac{84}{756} to lowest terms by extracting and canceling out 84.

$$x=\frac{1}{9}$$

The equation is now solved.

$$x=\frac{5}{42} x=\frac{1}{9}$$

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 3x.

$$18x\left(-18x+5\right)^{-1}\times 2+1=21x$$

Multiply 18 and 2 to get 36.

$$36x\left(-18x+5\right)^{-1}+1=21x$$

Subtract 21x from both sides.

$$36x\left(-18x+5\right)^{-1}+1-21x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$36x\left(-18x+5\right)^{-1}-21x=-1$$

Reorder the terms.

$$-21x+36\times \frac{1}{-18x+5}x=-1$$

Variable x cannot be equal to \frac{5}{18} since division by zero is not defined. Multiply both sides of the equation by -18x+5.

$$-21x\left(-18x+5\right)+36\times 1x=-\left(-18x+5\right)$$

Use the distributive property to multiply -21x by -18x+5.

$$378x^{2}-105x+36\times 1x=-\left(-18x+5\right)$$

Multiply 36 and 1 to get 36.

$$378x^{2}-105x+36x=-\left(-18x+5\right)$$

Combine -105x and 36x to get -69x.

$$378x^{2}-69x=-\left(-18x+5\right)$$

To find the opposite of -18x+5, find the opposite of each term.

$$378x^{2}-69x=18x-5$$

Subtract 18x from both sides.

$$378x^{2}-69x-18x=-5$$

Combine -69x and -18x to get -87x.

$$378x^{2}-87x=-5$$

Divide both sides by 378.

$$\frac{378x^{2}-87x}{378}=-\frac{5}{378}$$

Dividing by 378 undoes the multiplication by 378.

$$x^{2}+\left(-\frac{87}{378}\right)x=-\frac{5}{378}$$

Reduce the fraction \frac{-87}{378} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{29}{126}x=-\frac{5}{378}$$

Divide -\frac{29}{126}, the coefficient of the x term, by 2 to get -\frac{29}{252}. Then add the square of -\frac{29}{252} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{126}x+\left(-\frac{29}{252}\right)^{2}=-\frac{5}{378}+\left(-\frac{29}{252}\right)^{2}$$

Square -\frac{29}{252} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{126}x+\frac{841}{63504}=-\frac{5}{378}+\frac{841}{63504}$$

Add -\frac{5}{378} to \frac{841}{63504} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{29}{126}x+\frac{841}{63504}=\frac{1}{63504}$$

Factor x^{2}-\frac{29}{126}x+\frac{841}{63504}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{252}\right)^{2}=\frac{1}{63504}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{252}\right)^{2}}=\sqrt{\frac{1}{63504}}$$

Simplify.

$$x-\frac{29}{252}=\frac{1}{252} x-\frac{29}{252}=-\frac{1}{252}$$

Add \frac{29}{252} to both sides of the equation.

$$x=\frac{5}{42} x=\frac{1}{9}$$