Solve x^2+4x/7x-2-12-42x/x^2+4x=7

$$\frac { x ^ { 2 } + 4 x } { 7 x - 2 } - \frac { 12 - 42 x } { x ^ { 2 } + 4 x } = 7$$

Solve for x

$$x=1$$

$$x=2$$

$$x=19-\sqrt{349}\approx 0.318458308$$

$$x=\sqrt{349}+19\approx 37.681541692$$

Solution Steps

Variable x cannot be equal to any of the values -4,0,\frac{2}{7} since division by zero is not defined. Multiply both sides of the equation by x\left(7x-2\right)\left(x+4\right), the least common multiple of 7x-2,x^{2}+4x.

$$x\left(x+4\right)\left(x^{2}+4x\right)-\left(7x-2\right)\left(12-42x\right)=7x\left(7x-2\right)\left(x+4\right)$$

Use the distributive property to multiply x by x+4.

$$\left(x^{2}+4x\right)\left(x^{2}+4x\right)-\left(7x-2\right)\left(12-42x\right)=7x\left(7x-2\right)\left(x+4\right)$$

Multiply x^{2}+4x and x^{2}+4x to get \left(x^{2}+4x\right)^{2}.

$$\left(x^{2}+4x\right)^{2}-\left(7x-2\right)\left(12-42x\right)=7x\left(7x-2\right)\left(x+4\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+4x\right)^{2}.

$$\left(x^{2}\right)^{2}+8x^{2}x+16x^{2}-\left(7x-2\right)\left(12-42x\right)=7x\left(7x-2\right)\left(x+4\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}+8x^{2}x+16x^{2}-\left(7x-2\right)\left(12-42x\right)=7x\left(7x-2\right)\left(x+4\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{4}+8x^{3}+16x^{2}-\left(7x-2\right)\left(12-42x\right)=7x\left(7x-2\right)\left(x+4\right)$$

Use the distributive property to multiply 7x-2 by 12-42x and combine like terms.

$$x^{4}+8x^{3}+16x^{2}-\left(168x-294x^{2}-24\right)=7x\left(7x-2\right)\left(x+4\right)$$

To find the opposite of 168x-294x^{2}-24, find the opposite of each term.

$$x^{4}+8x^{3}+16x^{2}-168x+294x^{2}+24=7x\left(7x-2\right)\left(x+4\right)$$

Combine 16x^{2} and 294x^{2} to get 310x^{2}.

$$x^{4}+8x^{3}+310x^{2}-168x+24=7x\left(7x-2\right)\left(x+4\right)$$

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

$$x^{4}+8x^{3}+310x^{2}-168x+24=\left(49x^{2}-14x\right)\left(x+4\right)$$

Use the distributive property to multiply 49x^{2}-14x by x+4 and combine like terms.

$$x^{4}+8x^{3}+310x^{2}-168x+24=49x^{3}+182x^{2}-56x$$

Subtract 49x^{3} from both sides.

$$x^{4}+8x^{3}+310x^{2}-168x+24-49x^{3}=182x^{2}-56x$$

Combine 8x^{3} and -49x^{3} to get -41x^{3}.

$$x^{4}-41x^{3}+310x^{2}-168x+24=182x^{2}-56x$$

Subtract 182x^{2} from both sides.

$$x^{4}-41x^{3}+310x^{2}-168x+24-182x^{2}=-56x$$

Combine 310x^{2} and -182x^{2} to get 128x^{2}.

$$x^{4}-41x^{3}+128x^{2}-168x+24=-56x$$

Add 56x to both sides.

$$x^{4}-41x^{3}+128x^{2}-168x+24+56x=0$$

Combine -168x and 56x to get -112x.

$$x^{4}-41x^{3}+128x^{2}-112x+24=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 24 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±24,±12,±8,±6,±4,±3,±2,±1$$

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}-41x^{3}+128x^{2}-112x+24 by x-1 to get x^{3}-40x^{2}+88x-24. Solve the equation where the result equals to 0.

$$x^{3}-40x^{2}+88x-24=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -24 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±24,±12,±8,±6,±4,±3,±2,±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-40x^{2}+88x-24 by x-2 to get x^{2}-38x+12. Solve the equation where the result equals to 0.

$$x^{2}-38x+12=0$$

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, -38 for b, and 12 for c in the quadratic formula.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\times 1\times 12}}{2}$$

Do the calculations.

$$x=\frac{38±2\sqrt{349}}{2}$$

Solve the equation x^{2}-38x+12=0 when ± is plus and when ± is minus.

$$x=19-\sqrt{349} x=\sqrt{349}+19$$

List all found solutions.

$$x=1 x=2 x=19-\sqrt{349} x=\sqrt{349}+19$$