Solve 1/x^2+x-2+1/x^2+7x+10+1/x^2+13x+40=1/3x-3-frac{1}{21}

$$\frac { 1 } { x ^ { 2 } + x - 2 } + \frac { 1 } { x ^ { 2 } + 7 x + 10 } + \frac { 1 } { x ^ { 2 } + 13 x + 40 } = \frac { 1 } { 3 x - 3 } - \frac { 1 } { 21 }$$

Solve for x

$$x=-1$$

Solution Steps

Variable x cannot be equal to any of the values -8,-5,-2,1 since division by zero is not defined. Multiply both sides of the equation by 21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right), the least common multiple of x^{2}+x-2,x^{2}+7x+10,x^{2}+13x+40,3x-3,21.

$$21\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 21 by x+5.

$$\left(21x+105\right)\left(x+8\right)+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 21x+105 by x+8 and combine like terms.

$$21x^{2}+273x+840+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 21 by x-1.

$$21x^{2}+273x+840+\left(21x-21\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 21x-21 by x+8 and combine like terms.

$$21x^{2}+273x+840+21x^{2}+147x-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Combine 21x^{2} and 21x^{2} to get 42x^{2}.

$$42x^{2}+273x+840+147x-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Combine 273x and 147x to get 420x.

$$42x^{2}+420x+840-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Subtract 168 from 840 to get 672.

$$42x^{2}+420x+672+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 21 by x+2.

$$42x^{2}+420x+672+\left(21x+42\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 21x+42 by x-1 and combine like terms.

$$42x^{2}+420x+672+21x^{2}+21x-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Combine 42x^{2} and 21x^{2} to get 63x^{2}.

$$63x^{2}+420x+672+21x-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Combine 420x and 21x to get 441x.

$$63x^{2}+441x+672-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Subtract 42 from 672 to get 630.

$$63x^{2}+441x+630=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 7 by x+2.

$$63x^{2}+441x+630=\left(7x+14\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 7x+14 by x+5 and combine like terms.

$$63x^{2}+441x+630=\left(7x^{2}+49x+70\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Use the distributive property to multiply 7x^{2}+49x+70 by x+8 and combine like terms.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Multiply 21 and -\frac{1}{21} to get -1.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560-\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)$$

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

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x+1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)$$

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

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x^{2}-x+2\right)\left(x+5\right)\left(x+8\right)$$

Use the distributive property to multiply -x^{2}-x+2 by x+5 and combine like terms.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x^{3}-6x^{2}-3x+10\right)\left(x+8\right)$$

Use the distributive property to multiply -x^{3}-6x^{2}-3x+10 by x+8 and combine like terms.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560-x^{4}-14x^{3}-51x^{2}-14x+80$$

Combine 7x^{3} and -14x^{3} to get -7x^{3}.

$$63x^{2}+441x+630=-7x^{3}+105x^{2}+462x+560-x^{4}-51x^{2}-14x+80$$

Combine 105x^{2} and -51x^{2} to get 54x^{2}.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+462x+560-x^{4}-14x+80$$

Combine 462x and -14x to get 448x.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+448x+560-x^{4}+80$$

Add 560 and 80 to get 640.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+448x+640-x^{4}$$

Add 7x^{3} to both sides.

$$63x^{2}+441x+630+7x^{3}=54x^{2}+448x+640-x^{4}$$

Subtract 54x^{2} from both sides.

$$63x^{2}+441x+630+7x^{3}-54x^{2}=448x+640-x^{4}$$

Combine 63x^{2} and -54x^{2} to get 9x^{2}.

$$9x^{2}+441x+630+7x^{3}=448x+640-x^{4}$$

Subtract 448x from both sides.

$$9x^{2}+441x+630+7x^{3}-448x=640-x^{4}$$

Combine 441x and -448x to get -7x.

$$9x^{2}-7x+630+7x^{3}=640-x^{4}$$

Subtract 640 from both sides.

$$9x^{2}-7x+630+7x^{3}-640=-x^{4}$$

Subtract 640 from 630 to get -10.

$$9x^{2}-7x-10+7x^{3}=-x^{4}$$

Add x^{4} to both sides.

$$9x^{2}-7x-10+7x^{3}+x^{4}=0$$

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

$$x^{4}+7x^{3}+9x^{2}-7x-10=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -10 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±10,±5,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+7x^{3}+9x^{2}-7x-10 by x-1 to get x^{3}+8x^{2}+17x+10. Solve the equation where the result equals to 0.

$$x^{3}+8x^{2}+17x+10=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 10 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±10,±5,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+8x^{2}+17x+10 by x+1 to get x^{2}+7x+10. Solve the equation where the result equals to 0.

$$x^{2}+7x+10=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 7 for b, and 10 for c in the quadratic formula.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\times 10}}{2}$$

Do the calculations.

$$x=\frac{-7±3}{2}$$

Solve the equation x^{2}+7x+10=0 when ± is plus and when ± is minus.

$$x=-5 x=-2$$

Remove the values that the variable cannot be equal to.

$$x=-1$$

List all found solutions.

$$x=1 x=-1 x=-5 x=-2$$

Variable x cannot be equal to any of the values 1,-5,-2.

$$x=-1$$