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

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

Solve for a

$$a=3$$

$$a = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by a-4.

$$2\left(a-2\right)\left(a-4\right)=-\left(a-1\right)$$

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

$$\left(2a-4\right)\left(a-4\right)=-\left(a-1\right)$$

Use the distributive property to multiply 2a-4 by a-4 and combine like terms.

$$2a^{2}-12a+16=-\left(a-1\right)$$

To find the opposite of a-1, find the opposite of each term.

$$2a^{2}-12a+16=-a+1$$

Add a to both sides.

$$2a^{2}-12a+16+a=1$$

Combine -12a and a to get -11a.

$$2a^{2}-11a+16=1$$

Subtract 1 from both sides.

$$2a^{2}-11a+16-1=0$$

Subtract 1 from 16 to get 15.

$$2a^{2}-11a+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -11 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\times 15}}{2\times 2}$$

Square -11.

$$a=\frac{-\left(-11\right)±\sqrt{121-4\times 2\times 15}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-11\right)±\sqrt{121-8\times 15}}{2\times 2}$$

Multiply -8 times 15.

$$a=\frac{-\left(-11\right)±\sqrt{121-120}}{2\times 2}$$

Add 121 to -120.

$$a=\frac{-\left(-11\right)±\sqrt{1}}{2\times 2}$$

Take the square root of 1.

$$a=\frac{-\left(-11\right)±1}{2\times 2}$$

The opposite of -11 is 11.

$$a=\frac{11±1}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{11±1}{4}$$

Now solve the equation a=\frac{11±1}{4} when ± is plus. Add 11 to 1.

$$a=\frac{12}{4}$$

Divide 12 by 4.

$$a=3$$

Now solve the equation a=\frac{11±1}{4} when ± is minus. Subtract 1 from 11.

$$a=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$a=\frac{5}{2}$$

The equation is now solved.

$$a=3 a=\frac{5}{2}$$

Steps for Completing the Square

Variable a cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by a-4.

$$2\left(a-2\right)\left(a-4\right)=-\left(a-1\right)$$

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

$$\left(2a-4\right)\left(a-4\right)=-\left(a-1\right)$$

Use the distributive property to multiply 2a-4 by a-4 and combine like terms.

$$2a^{2}-12a+16=-\left(a-1\right)$$

To find the opposite of a-1, find the opposite of each term.

$$2a^{2}-12a+16=-a+1$$

Add a to both sides.

$$2a^{2}-12a+16+a=1$$

Combine -12a and a to get -11a.

$$2a^{2}-11a+16=1$$

Subtract 16 from both sides.

$$2a^{2}-11a=1-16$$

Subtract 16 from 1 to get -15.

$$2a^{2}-11a=-15$$

Divide both sides by 2.

$$\frac{2a^{2}-11a}{2}=-\frac{15}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}-\frac{11}{2}a=-\frac{15}{2}$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{11}{2}a+\frac{121}{16}=-\frac{15}{2}+\frac{121}{16}$$

Add -\frac{15}{2} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{11}{2}a+\frac{121}{16}=\frac{1}{16}$$

Factor a^{2}-\frac{11}{2}a+\frac{121}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{11}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{11}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$a-\frac{11}{4}=\frac{1}{4} a-\frac{11}{4}=-\frac{1}{4}$$

Add \frac{11}{4} to both sides of the equation.

$$a=3 a=\frac{5}{2}$$