Solve (2n-1)(2n-2)=12*(n-1)(n-2)

$$( 2 n - 1 ) ( 2 n - 2 ) = 12 \times ( n - 1 ) ( n - 2 )$$

Solve for n

$$n = \frac{11}{4} = 2\frac{3}{4} = 2.75$$

$$n=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2n-1 by 2n-2 and combine like terms.

$$4n^{2}-6n+2=12\left(n-1\right)\left(n-2\right)$$

Use the distributive property to multiply 12 by n-1.

$$4n^{2}-6n+2=\left(12n-12\right)\left(n-2\right)$$

Use the distributive property to multiply 12n-12 by n-2 and combine like terms.

$$4n^{2}-6n+2=12n^{2}-36n+24$$

Subtract 12n^{2} from both sides.

$$4n^{2}-6n+2-12n^{2}=-36n+24$$

Combine 4n^{2} and -12n^{2} to get -8n^{2}.

$$-8n^{2}-6n+2=-36n+24$$

Add 36n to both sides.

$$-8n^{2}-6n+2+36n=24$$

Combine -6n and 36n to get 30n.

$$-8n^{2}+30n+2=24$$

Subtract 24 from both sides.

$$-8n^{2}+30n+2-24=0$$

Subtract 24 from 2 to get -22.

$$-8n^{2}+30n-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 30 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-30±\sqrt{30^{2}-4\left(-8\right)\left(-22\right)}}{2\left(-8\right)}$$

Square 30.

$$n=\frac{-30±\sqrt{900-4\left(-8\right)\left(-22\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$n=\frac{-30±\sqrt{900+32\left(-22\right)}}{2\left(-8\right)}$$

Multiply 32 times -22.

$$n=\frac{-30±\sqrt{900-704}}{2\left(-8\right)}$$

Add 900 to -704.

$$n=\frac{-30±\sqrt{196}}{2\left(-8\right)}$$

Take the square root of 196.

$$n=\frac{-30±14}{2\left(-8\right)}$$

Multiply 2 times -8.

$$n=\frac{-30±14}{-16}$$

Now solve the equation n=\frac{-30±14}{-16} when ± is plus. Add -30 to 14.

$$n=-\frac{16}{-16}$$

Divide -16 by -16.

$$n=1$$

Now solve the equation n=\frac{-30±14}{-16} when ± is minus. Subtract 14 from -30.

$$n=-\frac{44}{-16}$$

Reduce the fraction \frac{-44}{-16} to lowest terms by extracting and canceling out 4.

$$n=\frac{11}{4}$$

The equation is now solved.

$$n=1 n=\frac{11}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2n-1 by 2n-2 and combine like terms.

$$4n^{2}-6n+2=12\left(n-1\right)\left(n-2\right)$$

Use the distributive property to multiply 12 by n-1.

$$4n^{2}-6n+2=\left(12n-12\right)\left(n-2\right)$$

Use the distributive property to multiply 12n-12 by n-2 and combine like terms.

$$4n^{2}-6n+2=12n^{2}-36n+24$$

Subtract 12n^{2} from both sides.

$$4n^{2}-6n+2-12n^{2}=-36n+24$$

Combine 4n^{2} and -12n^{2} to get -8n^{2}.

$$-8n^{2}-6n+2=-36n+24$$

Add 36n to both sides.

$$-8n^{2}-6n+2+36n=24$$

Combine -6n and 36n to get 30n.

$$-8n^{2}+30n+2=24$$

Subtract 2 from both sides.

$$-8n^{2}+30n=24-2$$

Subtract 2 from 24 to get 22.

$$-8n^{2}+30n=22$$

Divide both sides by -8.

$$\frac{-8n^{2}+30n}{-8}=\frac{22}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$n^{2}+\frac{30}{-8}n=\frac{22}{-8}$$

Reduce the fraction \frac{30}{-8} to lowest terms by extracting and canceling out 2.

$$n^{2}-\frac{15}{4}n=\frac{22}{-8}$$

Reduce the fraction \frac{22}{-8} to lowest terms by extracting and canceling out 2.

$$n^{2}-\frac{15}{4}n=-\frac{11}{4}$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{15}{4}n+\left(-\frac{15}{8}\right)^{2}=-\frac{11}{4}+\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{15}{4}n+\frac{225}{64}=-\frac{11}{4}+\frac{225}{64}$$

Add -\frac{11}{4} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{15}{4}n+\frac{225}{64}=\frac{49}{64}$$

Factor n^{2}-\frac{15}{4}n+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{15}{8}\right)^{2}=\frac{49}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{15}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Simplify.

$$n-\frac{15}{8}=\frac{7}{8} n-\frac{15}{8}=-\frac{7}{8}$$

Add \frac{15}{8} to both sides of the equation.

$$n=\frac{11}{4} n=1$$