Solve x-5/x-3+1/x=-7/x^2-3x

$$\frac { x - 5 } { x - 3 } + \frac { 1 } { x } = \frac { - 7 } { x ^ { 2 } - 3 x }$$

Solve for x

$$x=2$$

Steps Using Factoring

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x,x^{2}-3x.

$$x\left(x-5\right)+x-3=-7$$

Use the distributive property to multiply x by x-5.

$$x^{2}-5x+x-3=-7$$

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

$$x^{2}-4x-3=-7$$

Add 7 to both sides.

$$x^{2}-4x-3+7=0$$

Add -3 and 7 to get 4.

$$x^{2}-4x+4=0$$

To solve the equation, factor x^{2}-4x+4 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -4.

$$a=-2 b=-2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-2\right)\left(x-2\right)$$

Rewrite as a binomial square.

$$\left(x-2\right)^{2}$$

To find equation solution, solve x-2=0.

$$x=2$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x,x^{2}-3x.

$$x\left(x-5\right)+x-3=-7$$

Use the distributive property to multiply x by x-5.

$$x^{2}-5x+x-3=-7$$

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

$$x^{2}-4x-3=-7$$

Add 7 to both sides.

$$x^{2}-4x-3+7=0$$

Add -3 and 7 to get 4.

$$x^{2}-4x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -4.

$$a=-2 b=-2$$

Rewrite x^{2}-4x+4 as \left(x^{2}-2x\right)+\left(-2x+4\right).

$$\left(x^{2}-2x\right)+\left(-2x+4\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(x-2\right)-2\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x-2\right)$$

Rewrite as a binomial square.

$$\left(x-2\right)^{2}$$

To find equation solution, solve x-2=0.

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x,x^{2}-3x.

$$x\left(x-5\right)+x-3=-7$$

Use the distributive property to multiply x by x-5.

$$x^{2}-5x+x-3=-7$$

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

$$x^{2}-4x-3=-7$$

Add 7 to both sides.

$$x^{2}-4x-3+7=0$$

Add -3 and 7 to get 4.

$$x^{2}-4x+4=0$$

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

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

Square -4.

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

Multiply -4 times 4.

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

Add 16 to -16.

$$x=\frac{-\left(-4\right)±\sqrt{0}}{2}$$

Take the square root of 0.

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

The opposite of -4 is 4.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x,x^{2}-3x.

$$x\left(x-5\right)+x-3=-7$$

Use the distributive property to multiply x by x-5.

$$x^{2}-5x+x-3=-7$$

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

$$x^{2}-4x-3=-7$$

Add 3 to both sides.

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

Add -7 and 3 to get -4.

$$x^{2}-4x=-4$$

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.

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

Square -2.

$$x^{2}-4x+4=-4+4$$

Add -4 to 4.

$$x^{2}-4x+4=0$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-2=0 x-2=0$$

Add 2 to both sides of the equation.

$$x=2 x=2$$

The equation is now solved. Solutions are the same.

$$x=2$$