Solve 99/100=9n^2-9n+2/9n^2-1

$$\frac { 99 } { 100 } = \frac { 9 n ^ { 2 } - 9 n + 2 } { 9 n ^ { 2 } - 1 }$$

Solve for n

$$n = \frac{299}{3} = 99\frac{2}{3} \approx 99.666666667$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -\frac{1}{3},\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 100\left(3n-1\right)\left(3n+1\right), the least common multiple of 100,9n^{2}-1.

$$99\left(3n-1\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Use the distributive property to multiply 99 by 3n-1.

$$\left(297n-99\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Use the distributive property to multiply 297n-99 by 3n+1 and combine like terms.

$$891n^{2}-99=100\left(9n^{2}-9n+2\right)$$

Use the distributive property to multiply 100 by 9n^{2}-9n+2.

$$891n^{2}-99=900n^{2}-900n+200$$

Subtract 900n^{2} from both sides.

$$891n^{2}-99-900n^{2}=-900n+200$$

Combine 891n^{2} and -900n^{2} to get -9n^{2}.

$$-9n^{2}-99=-900n+200$$

Add 900n to both sides.

$$-9n^{2}-99+900n=200$$

Subtract 200 from both sides.

$$-9n^{2}-99+900n-200=0$$

Subtract 200 from -99 to get -299.

$$-9n^{2}-299+900n=0$$

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.

$$-9n^{2}+900n-299=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 900 for b, and -299 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-900±\sqrt{900^{2}-4\left(-9\right)\left(-299\right)}}{2\left(-9\right)}$$

Square 900.

$$n=\frac{-900±\sqrt{810000-4\left(-9\right)\left(-299\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$n=\frac{-900±\sqrt{810000+36\left(-299\right)}}{2\left(-9\right)}$$

Multiply 36 times -299.

$$n=\frac{-900±\sqrt{810000-10764}}{2\left(-9\right)}$$

Add 810000 to -10764.

$$n=\frac{-900±\sqrt{799236}}{2\left(-9\right)}$$

Take the square root of 799236.

$$n=\frac{-900±894}{2\left(-9\right)}$$

Multiply 2 times -9.

$$n=\frac{-900±894}{-18}$$

Now solve the equation n=\frac{-900±894}{-18} when ± is plus. Add -900 to 894.

$$n=-\frac{6}{-18}$$

Reduce the fraction \frac{-6}{-18} to lowest terms by extracting and canceling out 6.

$$n=\frac{1}{3}$$

Now solve the equation n=\frac{-900±894}{-18} when ± is minus. Subtract 894 from -900.

$$n=-\frac{1794}{-18}$$

Reduce the fraction \frac{-1794}{-18} to lowest terms by extracting and canceling out 6.

$$n=\frac{299}{3}$$

The equation is now solved.

$$n=\frac{1}{3} n=\frac{299}{3}$$

Variable n cannot be equal to \frac{1}{3}.

$$n=\frac{299}{3}$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -\frac{1}{3},\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 100\left(3n-1\right)\left(3n+1\right), the least common multiple of 100,9n^{2}-1.

$$99\left(3n-1\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Use the distributive property to multiply 99 by 3n-1.

$$\left(297n-99\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Use the distributive property to multiply 297n-99 by 3n+1 and combine like terms.

$$891n^{2}-99=100\left(9n^{2}-9n+2\right)$$

Use the distributive property to multiply 100 by 9n^{2}-9n+2.

$$891n^{2}-99=900n^{2}-900n+200$$

Subtract 900n^{2} from both sides.

$$891n^{2}-99-900n^{2}=-900n+200$$

Combine 891n^{2} and -900n^{2} to get -9n^{2}.

$$-9n^{2}-99=-900n+200$$

Add 900n to both sides.

$$-9n^{2}-99+900n=200$$

Add 99 to both sides.

$$-9n^{2}+900n=200+99$$

Add 200 and 99 to get 299.

$$-9n^{2}+900n=299$$

Divide both sides by -9.

$$\frac{-9n^{2}+900n}{-9}=\frac{299}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$n^{2}+\frac{900}{-9}n=\frac{299}{-9}$$

Divide 900 by -9.

$$n^{2}-100n=\frac{299}{-9}$$

Divide 299 by -9.

$$n^{2}-100n=-\frac{299}{9}$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-100n+\left(-50\right)^{2}=-\frac{299}{9}+\left(-50\right)^{2}$$

Square -50.

$$n^{2}-100n+2500=-\frac{299}{9}+2500$$

Add -\frac{299}{9} to 2500.

$$n^{2}-100n+2500=\frac{22201}{9}$$

Factor n^{2}-100n+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-50\right)^{2}=\frac{22201}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-50\right)^{2}}=\sqrt{\frac{22201}{9}}$$

Simplify.

$$n-50=\frac{149}{3} n-50=-\frac{149}{3}$$

Add 50 to both sides of the equation.

$$n=\frac{299}{3} n=\frac{1}{3}$$

Variable n cannot be equal to \frac{1}{3}.

$$n=\frac{299}{3}$$