Solve 1/x-5=3x/2x+28

$$\frac { 1 } { x - 5 } = \frac { 3 x } { 2 x + 28 }$$

Solve for x

$$x=7$$

$$x = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -14,5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-5\right)\left(x+14\right), the least common multiple of x-5,2x+28.

$$2x+28=\left(x-5\right)\times 3x$$

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

$$2x+28=\left(3x-15\right)x$$

Use the distributive property to multiply 3x-15 by x.

$$2x+28=3x^{2}-15x$$

Subtract 3x^{2} from both sides.

$$2x+28-3x^{2}=-15x$$

Add 15x to both sides.

$$2x+28-3x^{2}+15x=0$$

Combine 2x and 15x to get 17x.

$$17x+28-3x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-3x^{2}+17x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=17 ab=-3\times 28=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 17.

$$a=21 b=-4$$

Rewrite -3x^{2}+17x+28 as \left(-3x^{2}+21x\right)+\left(-4x+28\right).

$$\left(-3x^{2}+21x\right)+\left(-4x+28\right)$$

Factor out 3x in the first and 4 in the second group.

$$3x\left(-x+7\right)+4\left(-x+7\right)$$

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(3x+4\right)$$

To find equation solutions, solve -x+7=0 and 3x+4=0.

$$x=7 x=-\frac{4}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -14,5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-5\right)\left(x+14\right), the least common multiple of x-5,2x+28.

$$2x+28=\left(x-5\right)\times 3x$$

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

$$2x+28=\left(3x-15\right)x$$

Use the distributive property to multiply 3x-15 by x.

$$2x+28=3x^{2}-15x$$

Subtract 3x^{2} from both sides.

$$2x+28-3x^{2}=-15x$$

Add 15x to both sides.

$$2x+28-3x^{2}+15x=0$$

Combine 2x and 15x to get 17x.

$$17x+28-3x^{2}=0$$

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.

$$-3x^{2}+17x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 17 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-17±\sqrt{289+12\times 28}}{2\left(-3\right)}$$

Multiply 12 times 28.

$$x=\frac{-17±\sqrt{289+336}}{2\left(-3\right)}$$

Add 289 to 336.

$$x=\frac{-17±\sqrt{625}}{2\left(-3\right)}$$

Take the square root of 625.

$$x=\frac{-17±25}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-17±25}{-6}$$

Now solve the equation x=\frac{-17±25}{-6} when ± is plus. Add -17 to 25.

$$x=\frac{8}{-6}$$

Reduce the fraction \frac{8}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{4}{3}$$

Now solve the equation x=\frac{-17±25}{-6} when ± is minus. Subtract 25 from -17.

$$x=-\frac{42}{-6}$$

Divide -42 by -6.

$$x=7$$

The equation is now solved.

$$x=-\frac{4}{3} x=7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -14,5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-5\right)\left(x+14\right), the least common multiple of x-5,2x+28.

$$2x+28=\left(x-5\right)\times 3x$$

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

$$2x+28=\left(3x-15\right)x$$

Use the distributive property to multiply 3x-15 by x.

$$2x+28=3x^{2}-15x$$

Subtract 3x^{2} from both sides.

$$2x+28-3x^{2}=-15x$$

Add 15x to both sides.

$$2x+28-3x^{2}+15x=0$$

Combine 2x and 15x to get 17x.

$$17x+28-3x^{2}=0$$

Subtract 28 from both sides. Anything subtracted from zero gives its negation.

$$17x-3x^{2}=-28$$

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.

$$-3x^{2}+17x=-28$$

Divide both sides by -3.

$$\frac{-3x^{2}+17x}{-3}=-\frac{28}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{17}{-3}x=-\frac{28}{-3}$$

Divide 17 by -3.

$$x^{2}-\frac{17}{3}x=-\frac{28}{-3}$$

Divide -28 by -3.

$$x^{2}-\frac{17}{3}x=\frac{28}{3}$$

Divide -\frac{17}{3}, the coefficient of the x term, by 2 to get -\frac{17}{6}. Then add the square of -\frac{17}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{3}x+\left(-\frac{17}{6}\right)^{2}=\frac{28}{3}+\left(-\frac{17}{6}\right)^{2}$$

Square -\frac{17}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{3}x+\frac{289}{36}=\frac{28}{3}+\frac{289}{36}$$

Add \frac{28}{3} to \frac{289}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{17}{3}x+\frac{289}{36}=\frac{625}{36}$$

Factor x^{2}-\frac{17}{3}x+\frac{289}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{6}\right)^{2}=\frac{625}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{6}\right)^{2}}=\sqrt{\frac{625}{36}}$$

Simplify.

$$x-\frac{17}{6}=\frac{25}{6} x-\frac{17}{6}=-\frac{25}{6}$$

Add \frac{17}{6} to both sides of the equation.

$$x=7 x=-\frac{4}{3}$$