Solve 1/9x^2-6x+1+16/15x^2+25x-10=1/3x-1

$$\frac { 1 } { 9 x ^ { 2 } - 6 x + 1 } + \frac { 16 } { 15 x ^ { 2 } + 25 x - 10 } = \frac { 1 } { 3 x - 1 }$$

Solve for x

$$x=-\frac{2}{15}\approx -0.133333333$$

$$x=2$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 5\left(x+2\right)\left(3x-1\right)^{2}, the least common multiple of 9x^{2}-6x+1,15x^{2}+25x-10,3x-1.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by 16.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

Combine 5x and 48x to get 53x.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

Subtract 16 from 10 to get -6.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

Use the distributive property to multiply 5x+10 by 3x-1 and combine like terms.

$$53x-6=15x^{2}+25x-10$$

Subtract 15x^{2} from both sides.

$$53x-6-15x^{2}=25x-10$$

Subtract 25x from both sides.

$$53x-6-15x^{2}-25x=-10$$

Combine 53x and -25x to get 28x.

$$28x-6-15x^{2}=-10$$

Add 10 to both sides.

$$28x-6-15x^{2}+10=0$$

Add -6 and 10 to get 4.

$$28x+4-15x^{2}=0$$

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

$$-15x^{2}+28x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -15x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=28 ab=-15\times 4=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 28.

$$a=30 b=-2$$

Rewrite -15x^{2}+28x+4 as \left(-15x^{2}+30x\right)+\left(-2x+4\right).

$$\left(-15x^{2}+30x\right)+\left(-2x+4\right)$$

Factor out 15x in the first and 2 in the second group.

$$15x\left(-x+2\right)+2\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(15x+2\right)$$

To find equation solutions, solve -x+2=0 and 15x+2=0.

$$x=2 x=-\frac{2}{15}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 5\left(x+2\right)\left(3x-1\right)^{2}, the least common multiple of 9x^{2}-6x+1,15x^{2}+25x-10,3x-1.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by 16.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

Combine 5x and 48x to get 53x.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

Subtract 16 from 10 to get -6.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

Use the distributive property to multiply 5x+10 by 3x-1 and combine like terms.

$$53x-6=15x^{2}+25x-10$$

Subtract 15x^{2} from both sides.

$$53x-6-15x^{2}=25x-10$$

Subtract 25x from both sides.

$$53x-6-15x^{2}-25x=-10$$

Combine 53x and -25x to get 28x.

$$28x-6-15x^{2}=-10$$

Add 10 to both sides.

$$28x-6-15x^{2}+10=0$$

Add -6 and 10 to get 4.

$$28x+4-15x^{2}=0$$

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.

$$-15x^{2}+28x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 28 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-15\right)\times 4}}{2\left(-15\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-15\right)\times 4}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-28±\sqrt{784+60\times 4}}{2\left(-15\right)}$$

Multiply 60 times 4.

$$x=\frac{-28±\sqrt{784+240}}{2\left(-15\right)}$$

Add 784 to 240.

$$x=\frac{-28±\sqrt{1024}}{2\left(-15\right)}$$

Take the square root of 1024.

$$x=\frac{-28±32}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-28±32}{-30}$$

Now solve the equation x=\frac{-28±32}{-30} when ± is plus. Add -28 to 32.

$$x=\frac{4}{-30}$$

Reduce the fraction \frac{4}{-30} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{15}$$

Now solve the equation x=\frac{-28±32}{-30} when ± is minus. Subtract 32 from -28.

$$x=-\frac{60}{-30}$$

Divide -60 by -30.

$$x=2$$

The equation is now solved.

$$x=-\frac{2}{15} x=2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 5\left(x+2\right)\left(3x-1\right)^{2}, the least common multiple of 9x^{2}-6x+1,15x^{2}+25x-10,3x-1.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by 16.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

Combine 5x and 48x to get 53x.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

Subtract 16 from 10 to get -6.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

Use the distributive property to multiply 5x+10 by 3x-1 and combine like terms.

$$53x-6=15x^{2}+25x-10$$

Subtract 15x^{2} from both sides.

$$53x-6-15x^{2}=25x-10$$

Subtract 25x from both sides.

$$53x-6-15x^{2}-25x=-10$$

Combine 53x and -25x to get 28x.

$$28x-6-15x^{2}=-10$$

Add 6 to both sides.

$$28x-15x^{2}=-10+6$$

Add -10 and 6 to get -4.

$$28x-15x^{2}=-4$$

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.

$$-15x^{2}+28x=-4$$

Divide both sides by -15.

$$\frac{-15x^{2}+28x}{-15}=-\frac{4}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{28}{-15}x=-\frac{4}{-15}$$

Divide 28 by -15.

$$x^{2}-\frac{28}{15}x=-\frac{4}{-15}$$

Divide -4 by -15.

$$x^{2}-\frac{28}{15}x=\frac{4}{15}$$

Divide -\frac{28}{15}, the coefficient of the x term, by 2 to get -\frac{14}{15}. Then add the square of -\frac{14}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{15}x+\left(-\frac{14}{15}\right)^{2}=\frac{4}{15}+\left(-\frac{14}{15}\right)^{2}$$

Square -\frac{14}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{4}{15}+\frac{196}{225}$$

Add \frac{4}{15} to \frac{196}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{256}{225}$$

Factor x^{2}-\frac{28}{15}x+\frac{196}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{14}{15}\right)^{2}=\frac{256}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{15}\right)^{2}}=\sqrt{\frac{256}{225}}$$

Simplify.

$$x-\frac{14}{15}=\frac{16}{15} x-\frac{14}{15}=-\frac{16}{15}$$

Add \frac{14}{15} to both sides of the equation.

$$x=2 x=-\frac{2}{15}$$