Solve (a-a+25/a+1)=(a-1-4a-1/a+1)=

$$( a - a + \frac { 25 } { a + 1 } ) = ( a - 1 - \frac { 4 a - 1 } { a + 1 } ) =$$

Solve for a

$$a=\sqrt{29}+2\approx 7.385164807$$

$$a=2-\sqrt{29}\approx -3.385164807$$

Steps Using the Quadratic Formula

Variable a cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by a+1.

$$\left(a+1\right)a-a\left(a+1\right)+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$a^{2}+a-a\left(a+1\right)+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$a^{2}+a-a^{2}-a+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$a-a+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

Combine a and -a to get 0.

$$25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$25=a^{2}+a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$25=a^{2}+a-a-1-\left(4a-1\right)$$

Combine a and -a to get 0.

$$25=a^{2}-1-\left(4a-1\right)$$

To find the opposite of 4a-1, find the opposite of each term.

$$25=a^{2}-1-4a+1$$

Add -1 and 1 to get 0.

$$25=a^{2}-4a$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}-4a=25$$

Subtract 25 from both sides.

$$a^{2}-4a-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-25\right)}}{2}$$

Square -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$a=\frac{-\left(-4\right)±\sqrt{16+100}}{2}$$

Add 16 to 100.

$$a=\frac{-\left(-4\right)±\sqrt{116}}{2}$$

Take the square root of 116.

$$a=\frac{-\left(-4\right)±2\sqrt{29}}{2}$$

The opposite of -4 is 4.

$$a=\frac{4±2\sqrt{29}}{2}$$

Now solve the equation a=\frac{4±2\sqrt{29}}{2} when ± is plus. Add 4 to 2\sqrt{29}.

$$a=\frac{2\sqrt{29}+4}{2}$$

Divide 4+2\sqrt{29} by 2.

$$a=\sqrt{29}+2$$

Now solve the equation a=\frac{4±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from 4.

$$a=\frac{4-2\sqrt{29}}{2}$$

Divide 4-2\sqrt{29} by 2.

$$a=2-\sqrt{29}$$

The equation is now solved.

$$a=\sqrt{29}+2 a=2-\sqrt{29}$$

Steps for Completing the Square

Variable a cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by a+1.

$$\left(a+1\right)a-a\left(a+1\right)+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$a^{2}+a-a\left(a+1\right)+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$a^{2}+a-a^{2}-a+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$a-a+25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

Combine a and -a to get 0.

$$25=\left(a+1\right)a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$25=a^{2}+a+\left(a+1\right)\left(-1\right)-\left(4a-1\right)$$

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

$$25=a^{2}+a-a-1-\left(4a-1\right)$$

Combine a and -a to get 0.

$$25=a^{2}-1-\left(4a-1\right)$$

To find the opposite of 4a-1, find the opposite of each term.

$$25=a^{2}-1-4a+1$$

Add -1 and 1 to get 0.

$$25=a^{2}-4a$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}-4a=25$$

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

$$a^{2}-4a+\left(-2\right)^{2}=25+\left(-2\right)^{2}$$

Square -2.

$$a^{2}-4a+4=25+4$$

Add 25 to 4.

$$a^{2}-4a+4=29$$

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

$$\left(a-2\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{29}$$

Simplify.

$$a-2=\sqrt{29} a-2=-\sqrt{29}$$

Add 2 to both sides of the equation.

$$a=\sqrt{29}+2 a=2-\sqrt{29}$$