Solve 7/x-3-10/x-2-6/x-1=0

$$\frac { 7 } { x - 3 } - \frac { 10 } { x - 2 } - \frac { 6 } { x - 1 } = 0$$

Solve for x

$$x=4$$

$$x = \frac{13}{9} = 1\frac{4}{9} \approx 1.444444444$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 1,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x-1\right), the least common multiple of x-3,x-2,x-1.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x^{2}-3x+2 by 7.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x^{2}-4x+3 by 10.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

To find the opposite of 10x^{2}-40x+30, find the opposite of each term.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Combine 7x^{2} and -10x^{2} to get -3x^{2}.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Combine -21x and 40x to get 19x.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Subtract 30 from 14 to get -16.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

Use the distributive property to multiply x^{2}-5x+6 by 6.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

To find the opposite of 6x^{2}-30x+36, find the opposite of each term.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$-9x^{2}+19x-16+30x-36=0$$

Combine 19x and 30x to get 49x.

$$-9x^{2}+49x-16-36=0$$

Subtract 36 from -16 to get -52.

$$-9x^{2}+49x-52=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-52. To find a and b, set up a system to be solved.

$$a+b=49 ab=-9\left(-52\right)=468$$

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 468.

$$1,468 2,234 3,156 4,117 6,78 9,52 12,39 13,36 18,26$$

Calculate the sum for each pair.

$$1+468=469 2+234=236 3+156=159 4+117=121 6+78=84 9+52=61 12+39=51 13+36=49 18+26=44$$

The solution is the pair that gives sum 49.

$$a=36 b=13$$

Rewrite -9x^{2}+49x-52 as \left(-9x^{2}+36x\right)+\left(13x-52\right).

$$\left(-9x^{2}+36x\right)+\left(13x-52\right)$$

Factor out 9x in the first and -13 in the second group.

$$9x\left(-x+4\right)-13\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(9x-13\right)$$

To find equation solutions, solve -x+4=0 and 9x-13=0.

$$x=4 x=\frac{13}{9}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x-1\right), the least common multiple of x-3,x-2,x-1.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x^{2}-3x+2 by 7.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x^{2}-4x+3 by 10.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

To find the opposite of 10x^{2}-40x+30, find the opposite of each term.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Combine 7x^{2} and -10x^{2} to get -3x^{2}.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Combine -21x and 40x to get 19x.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Subtract 30 from 14 to get -16.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

Use the distributive property to multiply x^{2}-5x+6 by 6.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

To find the opposite of 6x^{2}-30x+36, find the opposite of each term.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$-9x^{2}+19x-16+30x-36=0$$

Combine 19x and 30x to get 49x.

$$-9x^{2}+49x-16-36=0$$

Subtract 36 from -16 to get -52.

$$-9x^{2}+49x-52=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 49 for b, and -52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-49±\sqrt{49^{2}-4\left(-9\right)\left(-52\right)}}{2\left(-9\right)}$$

Square 49.

$$x=\frac{-49±\sqrt{2401-4\left(-9\right)\left(-52\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-49±\sqrt{2401+36\left(-52\right)}}{2\left(-9\right)}$$

Multiply 36 times -52.

$$x=\frac{-49±\sqrt{2401-1872}}{2\left(-9\right)}$$

Add 2401 to -1872.

$$x=\frac{-49±\sqrt{529}}{2\left(-9\right)}$$

Take the square root of 529.

$$x=\frac{-49±23}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-49±23}{-18}$$

Now solve the equation x=\frac{-49±23}{-18} when ± is plus. Add -49 to 23.

$$x=-\frac{26}{-18}$$

Reduce the fraction \frac{-26}{-18} to lowest terms by extracting and canceling out 2.

$$x=\frac{13}{9}$$

Now solve the equation x=\frac{-49±23}{-18} when ± is minus. Subtract 23 from -49.

$$x=-\frac{72}{-18}$$

Divide -72 by -18.

$$x=4$$

The equation is now solved.

$$x=\frac{13}{9} x=4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x-1\right), the least common multiple of x-3,x-2,x-1.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x^{2}-3x+2 by 7.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Use the distributive property to multiply x^{2}-4x+3 by 10.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

To find the opposite of 10x^{2}-40x+30, find the opposite of each term.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Combine 7x^{2} and -10x^{2} to get -3x^{2}.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Combine -21x and 40x to get 19x.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Subtract 30 from 14 to get -16.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

Use the distributive property to multiply x^{2}-5x+6 by 6.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

To find the opposite of 6x^{2}-30x+36, find the opposite of each term.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$-9x^{2}+19x-16+30x-36=0$$

Combine 19x and 30x to get 49x.

$$-9x^{2}+49x-16-36=0$$

Subtract 36 from -16 to get -52.

$$-9x^{2}+49x-52=0$$

Add 52 to both sides. Anything plus zero gives itself.

$$-9x^{2}+49x=52$$

Divide both sides by -9.

$$\frac{-9x^{2}+49x}{-9}=\frac{52}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{49}{-9}x=\frac{52}{-9}$$

Divide 49 by -9.

$$x^{2}-\frac{49}{9}x=\frac{52}{-9}$$

Divide 52 by -9.

$$x^{2}-\frac{49}{9}x=-\frac{52}{9}$$

Divide -\frac{49}{9}, the coefficient of the x term, by 2 to get -\frac{49}{18}. Then add the square of -\frac{49}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{49}{9}x+\left(-\frac{49}{18}\right)^{2}=-\frac{52}{9}+\left(-\frac{49}{18}\right)^{2}$$

Square -\frac{49}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{49}{9}x+\frac{2401}{324}=-\frac{52}{9}+\frac{2401}{324}$$

Add -\frac{52}{9} to \frac{2401}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{49}{9}x+\frac{2401}{324}=\frac{529}{324}$$

Factor x^{2}-\frac{49}{9}x+\frac{2401}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{49}{18}\right)^{2}=\frac{529}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{49}{18}\right)^{2}}=\sqrt{\frac{529}{324}}$$

Simplify.

$$x-\frac{49}{18}=\frac{23}{18} x-\frac{49}{18}=-\frac{23}{18}$$

Add \frac{49}{18} to both sides of the equation.

$$x=4 x=\frac{13}{9}$$