Solve left(4-xright)left(x+5right)/6-left(2-xright)left(x+1right)/4=left(3-xright)left(x+2right)/8-(x+1)(x+2)/6+frac{83-x}{24}

$$\frac{ \left( 4-x \right) \left( x+5 \right) }{ 6 } - \frac{ \left( 2-x \right) \left( x+1 \right) }{ 4 } = \frac{ \left( 3-x \right) \left( x+2 \right) }{ 8 } - \frac{ (x+1)(x+2) }{ 6 } + \frac{ 83-x }{ 24 }$$

Solve for x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps by Finding Square Root

Multiply both sides of the equation by 24, the least common multiple of 6,4,8,24.

$$4\left(4-x\right)\left(x+5\right)-6\left(2-x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

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

$$\left(16-4x\right)\left(x+5\right)-6\left(2-x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply 16-4x by x+5 and combine like terms.

$$-4x+80-4x^{2}-6\left(2-x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -6 by 2-x.

$$-4x+80-4x^{2}+\left(-12+6x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -12+6x by x+1 and combine like terms.

$$-4x+80-4x^{2}-6x-12+6x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Combine -4x and -6x to get -10x.

$$-10x+80-4x^{2}-12+6x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Subtract 12 from 80 to get 68.

$$-10x+68-4x^{2}+6x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Combine -4x^{2} and 6x^{2} to get 2x^{2}.

$$-10x+68+2x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

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

$$-10x+68+2x^{2}=\left(9-3x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply 9-3x by x+2 and combine like terms.

$$-10x+68+2x^{2}=3x+18-3x^{2}-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -4 by x+1.

$$-10x+68+2x^{2}=3x+18-3x^{2}+\left(-4x-4\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -4x-4 by x+2 and combine like terms.

$$-10x+68+2x^{2}=3x+18-3x^{2}-4x^{2}-12x-8+83-x$$

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

$$-10x+68+2x^{2}=3x+18-7x^{2}-12x-8+83-x$$

Combine 3x and -12x to get -9x.

$$-10x+68+2x^{2}=-9x+18-7x^{2}-8+83-x$$

Subtract 8 from 18 to get 10.

$$-10x+68+2x^{2}=-9x+10-7x^{2}+83-x$$

Add 10 and 83 to get 93.

$$-10x+68+2x^{2}=-9x+93-7x^{2}-x$$

Combine -9x and -x to get -10x.

$$-10x+68+2x^{2}=-10x+93-7x^{2}$$

Add 10x to both sides.

$$-10x+68+2x^{2}+10x=93-7x^{2}$$

Combine -10x and 10x to get 0.

$$68+2x^{2}=93-7x^{2}$$

Add 7x^{2} to both sides.

$$68+2x^{2}+7x^{2}=93$$

Combine 2x^{2} and 7x^{2} to get 9x^{2}.

$$68+9x^{2}=93$$

Subtract 68 from both sides.

$$9x^{2}=93-68$$

Subtract 68 from 93 to get 25.

$$9x^{2}=25$$

Divide both sides by 9.

$$x^{2}=\frac{25}{9}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Multiply both sides of the equation by 24, the least common multiple of 6,4,8,24.

$$4\left(4-x\right)\left(x+5\right)-6\left(2-x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

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

$$\left(16-4x\right)\left(x+5\right)-6\left(2-x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply 16-4x by x+5 and combine like terms.

$$-4x+80-4x^{2}-6\left(2-x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -6 by 2-x.

$$-4x+80-4x^{2}+\left(-12+6x\right)\left(x+1\right)=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -12+6x by x+1 and combine like terms.

$$-4x+80-4x^{2}-6x-12+6x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Combine -4x and -6x to get -10x.

$$-10x+80-4x^{2}-12+6x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Subtract 12 from 80 to get 68.

$$-10x+68-4x^{2}+6x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Combine -4x^{2} and 6x^{2} to get 2x^{2}.

$$-10x+68+2x^{2}=3\left(3-x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

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

$$-10x+68+2x^{2}=\left(9-3x\right)\left(x+2\right)-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply 9-3x by x+2 and combine like terms.

$$-10x+68+2x^{2}=3x+18-3x^{2}-4\left(x+1\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -4 by x+1.

$$-10x+68+2x^{2}=3x+18-3x^{2}+\left(-4x-4\right)\left(x+2\right)+83-x$$

Use the distributive property to multiply -4x-4 by x+2 and combine like terms.

$$-10x+68+2x^{2}=3x+18-3x^{2}-4x^{2}-12x-8+83-x$$

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

$$-10x+68+2x^{2}=3x+18-7x^{2}-12x-8+83-x$$

Combine 3x and -12x to get -9x.

$$-10x+68+2x^{2}=-9x+18-7x^{2}-8+83-x$$

Subtract 8 from 18 to get 10.

$$-10x+68+2x^{2}=-9x+10-7x^{2}+83-x$$

Add 10 and 83 to get 93.

$$-10x+68+2x^{2}=-9x+93-7x^{2}-x$$

Combine -9x and -x to get -10x.

$$-10x+68+2x^{2}=-10x+93-7x^{2}$$

Add 10x to both sides.

$$-10x+68+2x^{2}+10x=93-7x^{2}$$

Combine -10x and 10x to get 0.

$$68+2x^{2}=93-7x^{2}$$

Subtract 93 from both sides.

$$68+2x^{2}-93=-7x^{2}$$

Subtract 93 from 68 to get -25.

$$-25+2x^{2}=-7x^{2}$$

Add 7x^{2} to both sides.

$$-25+2x^{2}+7x^{2}=0$$

Combine 2x^{2} and 7x^{2} to get 9x^{2}.

$$-25+9x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$9x^{2}-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 0 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 9\left(-25\right)}}{2\times 9}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 9\left(-25\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{0±\sqrt{-36\left(-25\right)}}{2\times 9}$$

Multiply -36 times -25.

$$x=\frac{0±\sqrt{900}}{2\times 9}$$

Take the square root of 900.

$$x=\frac{0±30}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{0±30}{18}$$

Now solve the equation x=\frac{0±30}{18} when ± is plus. Reduce the fraction \frac{30}{18} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{0±30}{18} when ± is minus. Reduce the fraction \frac{-30}{18} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

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