Solve u+2/u-4-1=u+1/u-3

$$\frac { u + 2 } { u - 4 } - 1 = \frac { u + 1 } { u - 3 }$$

Solve for u

$$u=2$$

$$u=7$$

Steps Using the Quadratic Formula

Variable u cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(u-4\right)\left(u-3\right), the least common multiple of u-4,u-3.

$$\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u-3 by u+2 and combine like terms.

$$u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u-4 by u-3 and combine like terms.

$$u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u^{2}-7u+12 by -1.

$$u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)$$

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

$$-u-6+7u-12=\left(u-4\right)\left(u+1\right)$$

Combine -u and 7u to get 6u.

$$6u-6-12=\left(u-4\right)\left(u+1\right)$$

Subtract 12 from -6 to get -18.

$$6u-18=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u-4 by u+1 and combine like terms.

$$6u-18=u^{2}-3u-4$$

Subtract u^{2} from both sides.

$$6u-18-u^{2}=-3u-4$$

Add 3u to both sides.

$$6u-18-u^{2}+3u=-4$$

Combine 6u and 3u to get 9u.

$$9u-18-u^{2}=-4$$

Add 4 to both sides.

$$9u-18-u^{2}+4=0$$

Add -18 and 4 to get -14.

$$9u-14-u^{2}=0$$

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.

$$-u^{2}+9u-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Square 9.

$$u=\frac{-9±\sqrt{81-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$u=\frac{-9±\sqrt{81+4\left(-14\right)}}{2\left(-1\right)}$$

Multiply 4 times -14.

$$u=\frac{-9±\sqrt{81-56}}{2\left(-1\right)}$$

Add 81 to -56.

$$u=\frac{-9±\sqrt{25}}{2\left(-1\right)}$$

Take the square root of 25.

$$u=\frac{-9±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$u=\frac{-9±5}{-2}$$

Now solve the equation u=\frac{-9±5}{-2} when ± is plus. Add -9 to 5.

$$u=-\frac{4}{-2}$$

Divide -4 by -2.

$$u=2$$

Now solve the equation u=\frac{-9±5}{-2} when ± is minus. Subtract 5 from -9.

$$u=-\frac{14}{-2}$$

Divide -14 by -2.

$$u=7$$

The equation is now solved.

$$u=2 u=7$$

Steps for Completing the Square

Variable u cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(u-4\right)\left(u-3\right), the least common multiple of u-4,u-3.

$$\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u-3 by u+2 and combine like terms.

$$u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u-4 by u-3 and combine like terms.

$$u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u^{2}-7u+12 by -1.

$$u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)$$

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

$$-u-6+7u-12=\left(u-4\right)\left(u+1\right)$$

Combine -u and 7u to get 6u.

$$6u-6-12=\left(u-4\right)\left(u+1\right)$$

Subtract 12 from -6 to get -18.

$$6u-18=\left(u-4\right)\left(u+1\right)$$

Use the distributive property to multiply u-4 by u+1 and combine like terms.

$$6u-18=u^{2}-3u-4$$

Subtract u^{2} from both sides.

$$6u-18-u^{2}=-3u-4$$

Add 3u to both sides.

$$6u-18-u^{2}+3u=-4$$

Combine 6u and 3u to get 9u.

$$9u-18-u^{2}=-4$$

Add 18 to both sides.

$$9u-u^{2}=-4+18$$

Add -4 and 18 to get 14.

$$9u-u^{2}=14$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-u^{2}+9u=14$$

Divide both sides by -1.

$$\frac{-u^{2}+9u}{-1}=\frac{14}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$u^{2}+\frac{9}{-1}u=\frac{14}{-1}$$

Divide 9 by -1.

$$u^{2}-9u=\frac{14}{-1}$$

Divide 14 by -1.

$$u^{2}-9u=-14$$

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

$$u^{2}-9u+\left(-\frac{9}{2}\right)^{2}=-14+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$u^{2}-9u+\frac{81}{4}=-14+\frac{81}{4}$$

Add -14 to \frac{81}{4}.

$$u^{2}-9u+\frac{81}{4}=\frac{25}{4}$$

Factor u^{2}-9u+\frac{81}{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}.

$$\left(u-\frac{9}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u-\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$u-\frac{9}{2}=\frac{5}{2} u-\frac{9}{2}=-\frac{5}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$u=7 u=2$$