Solve 2a-1/2a-2a/2a-1-1/2a-4a^2=(2a-1)(-2a+1)-2a(2a+1)-1/2a-4a^2

$$\frac { 2 a - 1 } { 2 a } - \frac { 2 a } { 2 a - 1 } - \frac { 1 } { 2 a - 4 a ^ { 2 } } = \frac { ( 2 a - 1 ) ( - 2 a + 1 ) - 2 a ( 2 a + 1 ) - 1 } { 2 a - 4 a ^ { 2 } }$$

Solve for a

$$a=-\frac{1}{4}=-0.25$$

Steps Using Factoring

Variable a cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2a\left(2a-1\right), the least common multiple of 2a,2a-1,2a-4a^{2}.

$$\left(2a-1\right)\left(2a-1\right)-2a\times 2a+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Multiply 2a-1 and 2a-1 to get \left(2a-1\right)^{2}.

$$\left(2a-1\right)^{2}-2a\times 2a+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Multiply 2a and 2a to get \left(2a\right)^{2}.

$$\left(2a-1\right)^{2}-\left(2a\right)^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2a-1\right)^{2}.

$$4a^{2}-4a+1-\left(2a\right)^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Expand \left(2a\right)^{2}.

$$4a^{2}-4a+1-2^{2}a^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-4a+1-4a^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Add 1 and 1 to get 2.

$$4a^{2}-4a+2-4a^{2}=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

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

$$4a^{2}-4a+2-4a^{2}=-\left(-4a^{2}+4a-1-2a\left(2a+1\right)-1\right)$$

To find the opposite of -4a^{2}+4a-1-2a\left(2a+1\right)-1, find the opposite of each term.

$$4a^{2}-4a+2-4a^{2}=-\left(-4a^{2}+4a-1-2a\left(2a+1\right)\right)+1$$

Add -4a^{2}+4a-1-2a\left(2a+1\right) to both sides.

$$4a^{2}-4a+2-4a^{2}+-4a^{2}+4a-1-2a\left(2a+1\right)=1$$

Subtract 1 from both sides.

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

Subtract 1 from 2 to get 1.

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

Multiply -1 and 4 to get -4.

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

Combine 4a^{2} and -4a^{2} to get 0.

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

Combine -4a and 4a to get 0.

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

Subtract 1 from 1 to get 0.

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

Use the distributive property to multiply -2a by 2a+1.

$$-4a^{2}-4a^{2}-2a=0$$

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

$$-8a^{2}-2a=0$$

Factor out a.

$$a\left(-8a-2\right)=0$$

To find equation solutions, solve a=0 and -8a-2=0.

$$a=0 a=-\frac{1}{4}$$

Variable a cannot be equal to 0.

$$a=-\frac{1}{4}$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2a\left(2a-1\right), the least common multiple of 2a,2a-1,2a-4a^{2}.

$$\left(2a-1\right)\left(2a-1\right)-2a\times 2a+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Multiply 2a-1 and 2a-1 to get \left(2a-1\right)^{2}.

$$\left(2a-1\right)^{2}-2a\times 2a+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Multiply 2a and 2a to get \left(2a\right)^{2}.

$$\left(2a-1\right)^{2}-\left(2a\right)^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2a-1\right)^{2}.

$$4a^{2}-4a+1-\left(2a\right)^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Expand \left(2a\right)^{2}.

$$4a^{2}-4a+1-2^{2}a^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-4a+1-4a^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Add 1 and 1 to get 2.

$$4a^{2}-4a+2-4a^{2}=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

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

$$4a^{2}-4a+2-4a^{2}=-\left(-4a^{2}+4a-1-2a\left(2a+1\right)-1\right)$$

To find the opposite of -4a^{2}+4a-1-2a\left(2a+1\right)-1, find the opposite of each term.

$$4a^{2}-4a+2-4a^{2}=-\left(-4a^{2}+4a-1-2a\left(2a+1\right)\right)+1$$

Add -4a^{2}+4a-1-2a\left(2a+1\right) to both sides.

$$4a^{2}-4a+2-4a^{2}+-4a^{2}+4a-1-2a\left(2a+1\right)=1$$

Subtract 1 from both sides.

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

Subtract 1 from 2 to get 1.

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

Multiply -1 and 4 to get -4.

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

Combine 4a^{2} and -4a^{2} to get 0.

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

Combine -4a and 4a to get 0.

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

Subtract 1 from 1 to get 0.

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

Use the distributive property to multiply -2a by 2a+1.

$$-4a^{2}-4a^{2}-2a=0$$

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

$$-8a^{2}-2a=0$$

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

$$a=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}}}{2\left(-8\right)}$$

Take the square root of \left(-2\right)^{2}.

$$a=\frac{-\left(-2\right)±2}{2\left(-8\right)}$$

The opposite of -2 is 2.

$$a=\frac{2±2}{2\left(-8\right)}$$

Multiply 2 times -8.

$$a=\frac{2±2}{-16}$$

Now solve the equation a=\frac{2±2}{-16} when ± is plus. Add 2 to 2.

$$a=\frac{4}{-16}$$

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

$$a=-\frac{1}{4}$$

Now solve the equation a=\frac{2±2}{-16} when ± is minus. Subtract 2 from 2.

$$a=\frac{0}{-16}$$

Divide 0 by -16.

$$a=0$$

The equation is now solved.

$$a=-\frac{1}{4} a=0$$

Variable a cannot be equal to 0.

$$a=-\frac{1}{4}$$

Steps for Completing the Square

Variable a cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2a\left(2a-1\right), the least common multiple of 2a,2a-1,2a-4a^{2}.

$$\left(2a-1\right)\left(2a-1\right)-2a\times 2a+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Multiply 2a-1 and 2a-1 to get \left(2a-1\right)^{2}.

$$\left(2a-1\right)^{2}-2a\times 2a+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Multiply 2a and 2a to get \left(2a\right)^{2}.

$$\left(2a-1\right)^{2}-\left(2a\right)^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2a-1\right)^{2}.

$$4a^{2}-4a+1-\left(2a\right)^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Expand \left(2a\right)^{2}.

$$4a^{2}-4a+1-2^{2}a^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-4a+1-4a^{2}+1=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

Add 1 and 1 to get 2.

$$4a^{2}-4a+2-4a^{2}=-\left(\left(2a-1\right)\left(-2a+1\right)-2a\left(2a+1\right)-1\right)$$

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

$$4a^{2}-4a+2-4a^{2}=-\left(-4a^{2}+4a-1-2a\left(2a+1\right)-1\right)$$

To find the opposite of -4a^{2}+4a-1-2a\left(2a+1\right)-1, find the opposite of each term.

$$4a^{2}-4a+2-4a^{2}=-\left(-4a^{2}+4a-1-2a\left(2a+1\right)\right)+1$$

Add -4a^{2}+4a-1-2a\left(2a+1\right) to both sides.

$$4a^{2}-4a+2-4a^{2}+-4a^{2}+4a-1-2a\left(2a+1\right)=1$$

Subtract 2 from both sides.

$$4a^{2}-4a-4a^{2}+-4a^{2}+4a-1-2a\left(2a+1\right)=1-2$$

Subtract 2 from 1 to get -1.

$$4a^{2}-4a-4a^{2}+-4a^{2}+4a-1-2a\left(2a+1\right)=-1$$

Multiply -1 and 4 to get -4.

$$4a^{2}-4a-4a^{2}-4a^{2}+4a-1-2a\left(2a+1\right)=-1$$

Combine 4a^{2} and -4a^{2} to get 0.

$$-4a-4a^{2}+4a-1-2a\left(2a+1\right)=-1$$

Combine -4a and 4a to get 0.

$$-4a^{2}-1-2a\left(2a+1\right)=-1$$

Use the distributive property to multiply -2a by 2a+1.

$$-4a^{2}-1-4a^{2}-2a=-1$$

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

$$-8a^{2}-1-2a=-1$$

Add 1 to both sides.

$$-8a^{2}-2a=-1+1$$

Add -1 and 1 to get 0.

$$-8a^{2}-2a=0$$

Divide both sides by -8.

$$\frac{-8a^{2}-2a}{-8}=\frac{0}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$a^{2}+\left(-\frac{2}{-8}\right)a=\frac{0}{-8}$$

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

$$a^{2}+\frac{1}{4}a=\frac{0}{-8}$$

Divide 0 by -8.

$$a^{2}+\frac{1}{4}a=0$$

Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{1}{4}a+\left(\frac{1}{8}\right)^{2}=\left(\frac{1}{8}\right)^{2}$$

Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{1}{4}a+\frac{1}{64}=\frac{1}{64}$$

Factor a^{2}+\frac{1}{4}a+\frac{1}{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}.

$$\left(a+\frac{1}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$a+\frac{1}{8}=\frac{1}{8} a+\frac{1}{8}=-\frac{1}{8}$$

Subtract \frac{1}{8} from both sides of the equation.

$$a=0 a=-\frac{1}{4}$$

Variable a cannot be equal to 0.

$$a=-\frac{1}{4}$$