Solve 14/36=frac{6}6+x^2

$$\frac { 14 } { 36 } = \frac { 6 } { 6 + x } ^ { 2 }$$

Solve for x

$$x=-\frac{18\sqrt{14}}{7}-6\approx -15.621404709$$

$$x=\frac{18\sqrt{14}}{7}-6\approx 3.621404709$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{14}{36} to lowest terms by extracting and canceling out 2.

$$\frac{7}{18}=\left(\frac{6}{6+x}\right)^{2}$$

To raise \frac{6}{6+x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{7}{18}=\frac{6^{2}}{\left(6+x\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{7}{18}=\frac{36}{\left(6+x\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+x\right)^{2}.

$$\frac{7}{18}=\frac{36}{36+12x+x^{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{36}{36+12x+x^{2}}=\frac{7}{18}$$

Subtract \frac{7}{18} from both sides.

$$\frac{36}{36+12x+x^{2}}-\frac{7}{18}=0$$

Factor 36+12x+x^{2}.

$$\frac{36}{\left(x+6\right)^{2}}-\frac{7}{18}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+6\right)^{2} and 18 is 18\left(x+6\right)^{2}. Multiply \frac{36}{\left(x+6\right)^{2}} times \frac{18}{18}. Multiply \frac{7}{18} times \frac{\left(x+6\right)^{2}}{\left(x+6\right)^{2}}.

$$\frac{36\times 18}{18\left(x+6\right)^{2}}-\frac{7\left(x+6\right)^{2}}{18\left(x+6\right)^{2}}=0$$

Since \frac{36\times 18}{18\left(x+6\right)^{2}} and \frac{7\left(x+6\right)^{2}}{18\left(x+6\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{36\times 18-7\left(x+6\right)^{2}}{18\left(x+6\right)^{2}}=0$$

Do the multiplications in 36\times 18-7\left(x+6\right)^{2}.

$$\frac{648-7x^{2}-84x-252}{18\left(x+6\right)^{2}}=0$$

Combine like terms in 648-7x^{2}-84x-252.

$$\frac{396-7x^{2}-84x}{18\left(x+6\right)^{2}}=0$$

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 18\left(x+6\right)^{2}.

$$396-7x^{2}-84x=0$$

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.

$$-7x^{2}-84x+396=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -84 for b, and 396 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\left(-7\right)\times 396}}{2\left(-7\right)}$$

Square -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\left(-7\right)\times 396}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-84\right)±\sqrt{7056+28\times 396}}{2\left(-7\right)}$$

Multiply 28 times 396.

$$x=\frac{-\left(-84\right)±\sqrt{7056+11088}}{2\left(-7\right)}$$

Add 7056 to 11088.

$$x=\frac{-\left(-84\right)±\sqrt{18144}}{2\left(-7\right)}$$

Take the square root of 18144.

$$x=\frac{-\left(-84\right)±36\sqrt{14}}{2\left(-7\right)}$$

The opposite of -84 is 84.

$$x=\frac{84±36\sqrt{14}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{84±36\sqrt{14}}{-14}$$

Now solve the equation x=\frac{84±36\sqrt{14}}{-14} when ± is plus. Add 84 to 36\sqrt{14}.

$$x=\frac{36\sqrt{14}+84}{-14}$$

Divide 84+36\sqrt{14} by -14.

$$x=-\frac{18\sqrt{14}}{7}-6$$

Now solve the equation x=\frac{84±36\sqrt{14}}{-14} when ± is minus. Subtract 36\sqrt{14} from 84.

$$x=\frac{84-36\sqrt{14}}{-14}$$

Divide 84-36\sqrt{14} by -14.

$$x=\frac{18\sqrt{14}}{7}-6$$

The equation is now solved.

$$x=-\frac{18\sqrt{14}}{7}-6 x=\frac{18\sqrt{14}}{7}-6$$

Steps for Completing the Square

Reduce the fraction \frac{14}{36} to lowest terms by extracting and canceling out 2.

$$\frac{7}{18}=\left(\frac{6}{6+x}\right)^{2}$$

To raise \frac{6}{6+x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{7}{18}=\frac{6^{2}}{\left(6+x\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{7}{18}=\frac{36}{\left(6+x\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+x\right)^{2}.

$$\frac{7}{18}=\frac{36}{36+12x+x^{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{36}{36+12x+x^{2}}=\frac{7}{18}$$

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 18\left(x+6\right)^{2}, the least common multiple of 36+12x+x^{2},18.

$$18\times 36=7\left(x+6\right)^{2}$$

Multiply 18 and 36 to get 648.

$$648=7\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$648=7\left(x^{2}+12x+36\right)$$

Use the distributive property to multiply 7 by x^{2}+12x+36.

$$648=7x^{2}+84x+252$$

Swap sides so that all variable terms are on the left hand side.

$$7x^{2}+84x+252=648$$

Subtract 252 from both sides.

$$7x^{2}+84x=648-252$$

Subtract 252 from 648 to get 396.

$$7x^{2}+84x=396$$

Divide both sides by 7.

$$\frac{7x^{2}+84x}{7}=\frac{396}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{84}{7}x=\frac{396}{7}$$

Divide 84 by 7.

$$x^{2}+12x=\frac{396}{7}$$

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

$$x^{2}+12x+6^{2}=\frac{396}{7}+6^{2}$$

Square 6.

$$x^{2}+12x+36=\frac{396}{7}+36$$

Add \frac{396}{7} to 36.

$$x^{2}+12x+36=\frac{648}{7}$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=\frac{648}{7}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{\frac{648}{7}}$$

Simplify.

$$x+6=\frac{18\sqrt{14}}{7} x+6=-\frac{18\sqrt{14}}{7}$$

Subtract 6 from both sides of the equation.

$$x=\frac{18\sqrt{14}}{7}-6 x=-\frac{18\sqrt{14}}{7}-6$$