Solve 8n^2-4(1-2n)(2+8n)=0

$$8 n ^ { 2 } - 4 ( 1 - 2 n ) ( 2 + 8 n ) = 0$$

Solve for n

$$n=\frac{\sqrt{10}+1}{9}\approx 0.462475296$$

$$n=\frac{1-\sqrt{10}}{9}\approx -0.240253073$$

Steps Using the Quadratic Formula

Multiply -1 and 4 to get -4.

$$8n^{2}-4\left(1-2n\right)\left(2+8n\right)=0$$

Use the distributive property to multiply -4 by 1-2n.

$$8n^{2}+\left(-4+8n\right)\left(2+8n\right)=0$$

Use the distributive property to multiply -4+8n by 2+8n and combine like terms.

$$8n^{2}-8-16n+64n^{2}=0$$

Combine 8n^{2} and 64n^{2} to get 72n^{2}.

$$72n^{2}-8-16n=0$$

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.

$$72n^{2}-16n-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 72 for a, -16 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 72\left(-8\right)}}{2\times 72}$$

Square -16.

$$n=\frac{-\left(-16\right)±\sqrt{256-4\times 72\left(-8\right)}}{2\times 72}$$

Multiply -4 times 72.

$$n=\frac{-\left(-16\right)±\sqrt{256-288\left(-8\right)}}{2\times 72}$$

Multiply -288 times -8.

$$n=\frac{-\left(-16\right)±\sqrt{256+2304}}{2\times 72}$$

Add 256 to 2304.

$$n=\frac{-\left(-16\right)±\sqrt{2560}}{2\times 72}$$

Take the square root of 2560.

$$n=\frac{-\left(-16\right)±16\sqrt{10}}{2\times 72}$$

The opposite of -16 is 16.

$$n=\frac{16±16\sqrt{10}}{2\times 72}$$

Multiply 2 times 72.

$$n=\frac{16±16\sqrt{10}}{144}$$

Now solve the equation n=\frac{16±16\sqrt{10}}{144} when ± is plus. Add 16 to 16\sqrt{10}.

$$n=\frac{16\sqrt{10}+16}{144}$$

Divide 16+16\sqrt{10} by 144.

$$n=\frac{\sqrt{10}+1}{9}$$

Now solve the equation n=\frac{16±16\sqrt{10}}{144} when ± is minus. Subtract 16\sqrt{10} from 16.

$$n=\frac{16-16\sqrt{10}}{144}$$

Divide 16-16\sqrt{10} by 144.

$$n=\frac{1-\sqrt{10}}{9}$$

The equation is now solved.

$$n=\frac{\sqrt{10}+1}{9} n=\frac{1-\sqrt{10}}{9}$$

Steps for Completing the Square

Multiply -1 and 4 to get -4.

$$8n^{2}-4\left(1-2n\right)\left(2+8n\right)=0$$

Use the distributive property to multiply -4 by 1-2n.

$$8n^{2}+\left(-4+8n\right)\left(2+8n\right)=0$$

Use the distributive property to multiply -4+8n by 2+8n and combine like terms.

$$8n^{2}-8-16n+64n^{2}=0$$

Combine 8n^{2} and 64n^{2} to get 72n^{2}.

$$72n^{2}-8-16n=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$72n^{2}-16n=8$$

Divide both sides by 72.

$$\frac{72n^{2}-16n}{72}=\frac{8}{72}$$

Dividing by 72 undoes the multiplication by 72.

$$n^{2}+\left(-\frac{16}{72}\right)n=\frac{8}{72}$$

Reduce the fraction \frac{-16}{72} to lowest terms by extracting and canceling out 8.

$$n^{2}-\frac{2}{9}n=\frac{8}{72}$$

Reduce the fraction \frac{8}{72} to lowest terms by extracting and canceling out 8.

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

Divide -\frac{2}{9}, the coefficient of the x term, by 2 to get -\frac{1}{9}. Then add the square of -\frac{1}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{9} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{2}{9}n+\frac{1}{81}=\frac{1}{9}+\frac{1}{81}$$

Add \frac{1}{9} to \frac{1}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{2}{9}n+\frac{1}{81}=\frac{10}{81}$$

Factor n^{2}-\frac{2}{9}n+\frac{1}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{1}{9}\right)^{2}=\frac{10}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{9}\right)^{2}}=\sqrt{\frac{10}{81}}$$

Simplify.

$$n-\frac{1}{9}=\frac{\sqrt{10}}{9} n-\frac{1}{9}=-\frac{\sqrt{10}}{9}$$

Add \frac{1}{9} to both sides of the equation.

$$n=\frac{\sqrt{10}+1}{9} n=\frac{1-\sqrt{10}}{9}$$