Solve 2x-3/x=x+3/x-5

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

Solve for x

$$x=1$$

$$x=15$$

Steps Using Factoring

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\left(2x-3\right)=x\left(x+3\right)$$

Use the distributive property to multiply x-5 by 2x-3 and combine like terms.

$$2x^{2}-13x+15=x\left(x+3\right)$$

Use the distributive property to multiply x by x+3.

$$2x^{2}-13x+15=x^{2}+3x$$

Subtract x^{2} from both sides.

$$2x^{2}-13x+15-x^{2}=3x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-13x+15=3x$$

Subtract 3x from both sides.

$$x^{2}-13x+15-3x=0$$

Combine -13x and -3x to get -16x.

$$x^{2}-16x+15=0$$

To solve the equation, factor x^{2}-16x+15 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.

$$a+b=-16 ab=15$$

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 15.

$$-1,-15 -3,-5$$

Calculate the sum for each pair.

$$-1-15=-16 -3-5=-8$$

The solution is the pair that gives sum -16.

$$a=-15 b=-1$$

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

$$\left(x-15\right)\left(x-1\right)$$

To find equation solutions, solve x-15=0 and x-1=0.

$$x=15 x=1$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\left(2x-3\right)=x\left(x+3\right)$$

Use the distributive property to multiply x-5 by 2x-3 and combine like terms.

$$2x^{2}-13x+15=x\left(x+3\right)$$

Use the distributive property to multiply x by x+3.

$$2x^{2}-13x+15=x^{2}+3x$$

Subtract x^{2} from both sides.

$$2x^{2}-13x+15-x^{2}=3x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-13x+15=3x$$

Subtract 3x from both sides.

$$x^{2}-13x+15-3x=0$$

Combine -13x and -3x to get -16x.

$$x^{2}-16x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 15=15$$

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 15.

$$-1,-15 -3,-5$$

Calculate the sum for each pair.

$$-1-15=-16 -3-5=-8$$

The solution is the pair that gives sum -16.

$$a=-15 b=-1$$

Rewrite x^{2}-16x+15 as \left(x^{2}-15x\right)+\left(-x+15\right).

$$\left(x^{2}-15x\right)+\left(-x+15\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-15\right)-\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x-1\right)$$

To find equation solutions, solve x-15=0 and x-1=0.

$$x=15 x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\left(2x-3\right)=x\left(x+3\right)$$

Use the distributive property to multiply x-5 by 2x-3 and combine like terms.

$$2x^{2}-13x+15=x\left(x+3\right)$$

Use the distributive property to multiply x by x+3.

$$2x^{2}-13x+15=x^{2}+3x$$

Subtract x^{2} from both sides.

$$2x^{2}-13x+15-x^{2}=3x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-13x+15=3x$$

Subtract 3x from both sides.

$$x^{2}-13x+15-3x=0$$

Combine -13x and -3x to get -16x.

$$x^{2}-16x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 15}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 15}}{2}$$

Multiply -4 times 15.

$$x=\frac{-\left(-16\right)±\sqrt{256-60}}{2}$$

Add 256 to -60.

$$x=\frac{-\left(-16\right)±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-\left(-16\right)±14}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±14}{2}$$

Now solve the equation x=\frac{16±14}{2} when ± is plus. Add 16 to 14.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Now solve the equation x=\frac{16±14}{2} when ± is minus. Subtract 14 from 16.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=15 x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\left(2x-3\right)=x\left(x+3\right)$$

Use the distributive property to multiply x-5 by 2x-3 and combine like terms.

$$2x^{2}-13x+15=x\left(x+3\right)$$

Use the distributive property to multiply x by x+3.

$$2x^{2}-13x+15=x^{2}+3x$$

Subtract x^{2} from both sides.

$$2x^{2}-13x+15-x^{2}=3x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-13x+15=3x$$

Subtract 3x from both sides.

$$x^{2}-13x+15-3x=0$$

Combine -13x and -3x to get -16x.

$$x^{2}-16x+15=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-16x=-15$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-15+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-15+64$$

Add -15 to 64.

$$x^{2}-16x+64=49$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-8=7 x-8=-7$$

Add 8 to both sides of the equation.

$$x=15 x=1$$