Solve 2a-3/a+2+a+2/a-1=a+2/a-1+3

$$\frac { 2 a - 3 } { a + 2 } + \frac { a + 2 } { a - 1 } = \frac { a + 2 } { a - 1 } + 3$$

Solve for a

$$a=-9$$

Steps Using the Quadratic Formula

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

$$\left(a-1\right)\left(2a-3\right)+\left(a+2\right)\left(a+2\right)=\left(a+2\right)\left(a+2\right)+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$\left(a-1\right)\left(2a-3\right)+\left(a+2\right)^{2}=\left(a+2\right)\left(a+2\right)+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$\left(a-1\right)\left(2a-3\right)+\left(a+2\right)^{2}=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Use the distributive property to multiply a-1 by 2a-3 and combine like terms.

$$2a^{2}-5a+3+\left(a+2\right)^{2}=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$2a^{2}-5a+3+a^{2}+4a+4=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Combine 2a^{2} and a^{2} to get 3a^{2}.

$$3a^{2}-5a+3+4a+4=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Combine -5a and 4a to get -a.

$$3a^{2}-a+3+4=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Add 3 and 4 to get 7.

$$3a^{2}-a+7=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$3a^{2}-a+7=a^{2}+4a+4+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$3a^{2}-a+7=a^{2}+4a+4+\left(a^{2}+a-2\right)\times 3$$

Use the distributive property to multiply a^{2}+a-2 by 3.

$$3a^{2}-a+7=a^{2}+4a+4+3a^{2}+3a-6$$

Combine a^{2} and 3a^{2} to get 4a^{2}.

$$3a^{2}-a+7=4a^{2}+4a+4+3a-6$$

Combine 4a and 3a to get 7a.

$$3a^{2}-a+7=4a^{2}+7a+4-6$$

Subtract 6 from 4 to get -2.

$$3a^{2}-a+7=4a^{2}+7a-2$$

Subtract 4a^{2} from both sides.

$$3a^{2}-a+7-4a^{2}=7a-2$$

Combine 3a^{2} and -4a^{2} to get -a^{2}.

$$-a^{2}-a+7=7a-2$$

Subtract 7a from both sides.

$$-a^{2}-a+7-7a=-2$$

Combine -a and -7a to get -8a.

$$-a^{2}-8a+7=-2$$

Add 2 to both sides.

$$-a^{2}-8a+7+2=0$$

Add 7 and 2 to get 9.

$$-a^{2}-8a+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Square -8.

$$a=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-8\right)±\sqrt{64+4\times 9}}{2\left(-1\right)}$$

Multiply 4 times 9.

$$a=\frac{-\left(-8\right)±\sqrt{64+36}}{2\left(-1\right)}$$

Add 64 to 36.

$$a=\frac{-\left(-8\right)±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$a=\frac{-\left(-8\right)±10}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$a=\frac{8±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{8±10}{-2}$$

Now solve the equation a=\frac{8±10}{-2} when ± is plus. Add 8 to 10.

$$a=\frac{18}{-2}$$

Divide 18 by -2.

$$a=-9$$

Now solve the equation a=\frac{8±10}{-2} when ± is minus. Subtract 10 from 8.

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

Divide -2 by -2.

$$a=1$$

The equation is now solved.

$$a=-9 a=1$$

Variable a cannot be equal to 1.

$$a=-9$$

Steps for Completing the Square

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

$$\left(a-1\right)\left(2a-3\right)+\left(a+2\right)\left(a+2\right)=\left(a+2\right)\left(a+2\right)+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$\left(a-1\right)\left(2a-3\right)+\left(a+2\right)^{2}=\left(a+2\right)\left(a+2\right)+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$\left(a-1\right)\left(2a-3\right)+\left(a+2\right)^{2}=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Use the distributive property to multiply a-1 by 2a-3 and combine like terms.

$$2a^{2}-5a+3+\left(a+2\right)^{2}=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$2a^{2}-5a+3+a^{2}+4a+4=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Combine 2a^{2} and a^{2} to get 3a^{2}.

$$3a^{2}-5a+3+4a+4=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Combine -5a and 4a to get -a.

$$3a^{2}-a+3+4=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

Add 3 and 4 to get 7.

$$3a^{2}-a+7=\left(a+2\right)^{2}+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$3a^{2}-a+7=a^{2}+4a+4+\left(a-1\right)\left(a+2\right)\times 3$$

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

$$3a^{2}-a+7=a^{2}+4a+4+\left(a^{2}+a-2\right)\times 3$$

Use the distributive property to multiply a^{2}+a-2 by 3.

$$3a^{2}-a+7=a^{2}+4a+4+3a^{2}+3a-6$$

Combine a^{2} and 3a^{2} to get 4a^{2}.

$$3a^{2}-a+7=4a^{2}+4a+4+3a-6$$

Combine 4a and 3a to get 7a.

$$3a^{2}-a+7=4a^{2}+7a+4-6$$

Subtract 6 from 4 to get -2.

$$3a^{2}-a+7=4a^{2}+7a-2$$

Subtract 4a^{2} from both sides.

$$3a^{2}-a+7-4a^{2}=7a-2$$

Combine 3a^{2} and -4a^{2} to get -a^{2}.

$$-a^{2}-a+7=7a-2$$

Subtract 7a from both sides.

$$-a^{2}-a+7-7a=-2$$

Combine -a and -7a to get -8a.

$$-a^{2}-8a+7=-2$$

Subtract 7 from both sides.

$$-a^{2}-8a=-2-7$$

Subtract 7 from -2 to get -9.

$$-a^{2}-8a=-9$$

Divide both sides by -1.

$$\frac{-a^{2}-8a}{-1}=-\frac{9}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\left(-\frac{8}{-1}\right)a=-\frac{9}{-1}$$

Divide -8 by -1.

$$a^{2}+8a=-\frac{9}{-1}$$

Divide -9 by -1.

$$a^{2}+8a=9$$

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

$$a^{2}+8a+4^{2}=9+4^{2}$$

Square 4.

$$a^{2}+8a+16=9+16$$

Add 9 to 16.

$$a^{2}+8a+16=25$$

Factor a^{2}+8a+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$a+4=5 a+4=-5$$

Subtract 4 from both sides of the equation.

$$a=1 a=-9$$

Variable a cannot be equal to 1.

$$a=-9$$