Solve 6/x-2+5/x=3

$$\frac { 6 } { x - 2 } + \frac { 5 } { x } = 3$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

$$x=5$$

Steps Using Factoring By Grouping

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

$$x\times 6+\left(x-2\right)\times 5=3x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 5.

$$x\times 6+5x-10=3x\left(x-2\right)$$

Combine x\times 6 and 5x to get 11x.

$$11x-10=3x\left(x-2\right)$$

Use the distributive property to multiply 3x by x-2.

$$11x-10=3x^{2}-6x$$

Subtract 3x^{2} from both sides.

$$11x-10-3x^{2}=-6x$$

Add 6x to both sides.

$$11x-10-3x^{2}+6x=0$$

Combine 11x and 6x to get 17x.

$$17x-10-3x^{2}=0$$

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

$$-3x^{2}+17x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=17 ab=-3\left(-10\right)=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 17.

$$a=15 b=2$$

Rewrite -3x^{2}+17x-10 as \left(-3x^{2}+15x\right)+\left(2x-10\right).

$$\left(-3x^{2}+15x\right)+\left(2x-10\right)$$

Factor out 3x in the first and -2 in the second group.

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

Factor out common term -x+5 by using distributive property.

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

To find equation solutions, solve -x+5=0 and 3x-2=0.

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

Steps Using the Quadratic Formula

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

$$x\times 6+\left(x-2\right)\times 5=3x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 5.

$$x\times 6+5x-10=3x\left(x-2\right)$$

Combine x\times 6 and 5x to get 11x.

$$11x-10=3x\left(x-2\right)$$

Use the distributive property to multiply 3x by x-2.

$$11x-10=3x^{2}-6x$$

Subtract 3x^{2} from both sides.

$$11x-10-3x^{2}=-6x$$

Add 6x to both sides.

$$11x-10-3x^{2}+6x=0$$

Combine 11x and 6x to get 17x.

$$17x-10-3x^{2}=0$$

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.

$$-3x^{2}+17x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 17 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-17±\sqrt{289+12\left(-10\right)}}{2\left(-3\right)}$$

Multiply 12 times -10.

$$x=\frac{-17±\sqrt{289-120}}{2\left(-3\right)}$$

Add 289 to -120.

$$x=\frac{-17±\sqrt{169}}{2\left(-3\right)}$$

Take the square root of 169.

$$x=\frac{-17±13}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-17±13}{-6}$$

Now solve the equation x=\frac{-17±13}{-6} when ± is plus. Add -17 to 13.

$$x=-\frac{4}{-6}$$

Reduce the fraction \frac{-4}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{-17±13}{-6} when ± is minus. Subtract 13 from -17.

$$x=-\frac{30}{-6}$$

Divide -30 by -6.

$$x=5$$

The equation is now solved.

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

Steps for Completing the Square

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

$$x\times 6+\left(x-2\right)\times 5=3x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 5.

$$x\times 6+5x-10=3x\left(x-2\right)$$

Combine x\times 6 and 5x to get 11x.

$$11x-10=3x\left(x-2\right)$$

Use the distributive property to multiply 3x by x-2.

$$11x-10=3x^{2}-6x$$

Subtract 3x^{2} from both sides.

$$11x-10-3x^{2}=-6x$$

Add 6x to both sides.

$$11x-10-3x^{2}+6x=0$$

Combine 11x and 6x to get 17x.

$$17x-10-3x^{2}=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$17x-3x^{2}=10$$

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.

$$-3x^{2}+17x=10$$

Divide both sides by -3.

$$\frac{-3x^{2}+17x}{-3}=\frac{10}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{17}{-3}x=\frac{10}{-3}$$

Divide 17 by -3.

$$x^{2}-\frac{17}{3}x=\frac{10}{-3}$$

Divide 10 by -3.

$$x^{2}-\frac{17}{3}x=-\frac{10}{3}$$

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.

$$x^{2}-\frac{17}{3}x+\left(-\frac{17}{6}\right)^{2}=-\frac{10}{3}+\left(-\frac{17}{6}\right)^{2}$$

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

$$x^{2}-\frac{17}{3}x+\frac{289}{36}=-\frac{10}{3}+\frac{289}{36}$$

Add -\frac{10}{3} to \frac{289}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{17}{3}x+\frac{289}{36}=\frac{169}{36}$$

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}.

$$\left(x-\frac{17}{6}\right)^{2}=\frac{169}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Simplify.

$$x-\frac{17}{6}=\frac{13}{6} x-\frac{17}{6}=-\frac{13}{6}$$

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

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