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

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

Solve for x

$$x=\frac{1}{7}\approx 0.142857143$$

$$x=7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{3}{5},\frac{5}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(3x-5\right)\left(5x-3\right), the least common multiple of 3-5x,5-3x,2.

$$\left(10-6x\right)\left(5-3x\right)+\left(6-10x\right)\left(3-5x\right)=5\left(3x-5\right)\left(5x-3\right)$$

Use the distributive property to multiply 10-6x by 5-3x and combine like terms.

$$50-60x+18x^{2}+\left(6-10x\right)\left(3-5x\right)=5\left(3x-5\right)\left(5x-3\right)$$

Use the distributive property to multiply 6-10x by 3-5x and combine like terms.

$$50-60x+18x^{2}+18-60x+50x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Add 50 and 18 to get 68.

$$68-60x+18x^{2}-60x+50x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Combine -60x and -60x to get -120x.

$$68-120x+18x^{2}+50x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Combine 18x^{2} and 50x^{2} to get 68x^{2}.

$$68-120x+68x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Use the distributive property to multiply 5 by 3x-5.

$$68-120x+68x^{2}=\left(15x-25\right)\left(5x-3\right)$$

Use the distributive property to multiply 15x-25 by 5x-3 and combine like terms.

$$68-120x+68x^{2}=75x^{2}-170x+75$$

Subtract 75x^{2} from both sides.

$$68-120x+68x^{2}-75x^{2}=-170x+75$$

Combine 68x^{2} and -75x^{2} to get -7x^{2}.

$$68-120x-7x^{2}=-170x+75$$

Add 170x to both sides.

$$68-120x-7x^{2}+170x=75$$

Combine -120x and 170x to get 50x.

$$68+50x-7x^{2}=75$$

Subtract 75 from both sides.

$$68+50x-7x^{2}-75=0$$

Subtract 75 from 68 to get -7.

$$-7+50x-7x^{2}=0$$

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.

$$-7x^{2}+50x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 50 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-7\right)\left(-7\right)}}{2\left(-7\right)}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-7\right)\left(-7\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-50±\sqrt{2500+28\left(-7\right)}}{2\left(-7\right)}$$

Multiply 28 times -7.

$$x=\frac{-50±\sqrt{2500-196}}{2\left(-7\right)}$$

Add 2500 to -196.

$$x=\frac{-50±\sqrt{2304}}{2\left(-7\right)}$$

Take the square root of 2304.

$$x=\frac{-50±48}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-50±48}{-14}$$

Now solve the equation x=\frac{-50±48}{-14} when ± is plus. Add -50 to 48.

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

Reduce the fraction \frac{-2}{-14} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{7}$$

Now solve the equation x=\frac{-50±48}{-14} when ± is minus. Subtract 48 from -50.

$$x=-\frac{98}{-14}$$

Divide -98 by -14.

$$x=7$$

The equation is now solved.

$$x=\frac{1}{7} x=7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{3}{5},\frac{5}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(3x-5\right)\left(5x-3\right), the least common multiple of 3-5x,5-3x,2.

$$\left(10-6x\right)\left(5-3x\right)+\left(6-10x\right)\left(3-5x\right)=5\left(3x-5\right)\left(5x-3\right)$$

Use the distributive property to multiply 10-6x by 5-3x and combine like terms.

$$50-60x+18x^{2}+\left(6-10x\right)\left(3-5x\right)=5\left(3x-5\right)\left(5x-3\right)$$

Use the distributive property to multiply 6-10x by 3-5x and combine like terms.

$$50-60x+18x^{2}+18-60x+50x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Add 50 and 18 to get 68.

$$68-60x+18x^{2}-60x+50x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Combine -60x and -60x to get -120x.

$$68-120x+18x^{2}+50x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Combine 18x^{2} and 50x^{2} to get 68x^{2}.

$$68-120x+68x^{2}=5\left(3x-5\right)\left(5x-3\right)$$

Use the distributive property to multiply 5 by 3x-5.

$$68-120x+68x^{2}=\left(15x-25\right)\left(5x-3\right)$$

Use the distributive property to multiply 15x-25 by 5x-3 and combine like terms.

$$68-120x+68x^{2}=75x^{2}-170x+75$$

Subtract 75x^{2} from both sides.

$$68-120x+68x^{2}-75x^{2}=-170x+75$$

Combine 68x^{2} and -75x^{2} to get -7x^{2}.

$$68-120x-7x^{2}=-170x+75$$

Add 170x to both sides.

$$68-120x-7x^{2}+170x=75$$

Combine -120x and 170x to get 50x.

$$68+50x-7x^{2}=75$$

Subtract 68 from both sides.

$$50x-7x^{2}=75-68$$

Subtract 68 from 75 to get 7.

$$50x-7x^{2}=7$$

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.

$$-7x^{2}+50x=7$$

Divide both sides by -7.

$$\frac{-7x^{2}+50x}{-7}=\frac{7}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{50}{-7}x=\frac{7}{-7}$$

Divide 50 by -7.

$$x^{2}-\frac{50}{7}x=\frac{7}{-7}$$

Divide 7 by -7.

$$x^{2}-\frac{50}{7}x=-1$$

Divide -\frac{50}{7}, the coefficient of the x term, by 2 to get -\frac{25}{7}. Then add the square of -\frac{25}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{50}{7}x+\left(-\frac{25}{7}\right)^{2}=-1+\left(-\frac{25}{7}\right)^{2}$$

Square -\frac{25}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{50}{7}x+\frac{625}{49}=-1+\frac{625}{49}$$

Add -1 to \frac{625}{49}.

$$x^{2}-\frac{50}{7}x+\frac{625}{49}=\frac{576}{49}$$

Factor x^{2}-\frac{50}{7}x+\frac{625}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{7}\right)^{2}=\frac{576}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{7}\right)^{2}}=\sqrt{\frac{576}{49}}$$

Simplify.

$$x-\frac{25}{7}=\frac{24}{7} x-\frac{25}{7}=-\frac{24}{7}$$

Add \frac{25}{7} to both sides of the equation.

$$x=7 x=\frac{1}{7}$$