Solve 8x/x^2+10x+25=4/x^2+7x+10-1/x+2

$$\frac { 8 x } { x ^ { 2 } + 10 x + 25 } = \frac { 4 } { x ^ { 2 } + 7 x + 10 } - \frac { 1 } { x + 2 }$$

Solve for x

$$x=\frac{2\sqrt{19}-11}{9}\approx -0.253578013$$

$$x=\frac{-2\sqrt{19}-11}{9}\approx -2.190866432$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+5\right)^{2}, the least common multiple of x^{2}+10x+25,x^{2}+7x+10,x+2.

$$\left(x+2\right)\times 8x=\left(x+5\right)\times 4-\left(x+5\right)^{2}$$

Use the distributive property to multiply x+2 by 8.

$$\left(8x+16\right)x=\left(x+5\right)\times 4-\left(x+5\right)^{2}$$

Use the distributive property to multiply 8x+16 by x.

$$8x^{2}+16x=\left(x+5\right)\times 4-\left(x+5\right)^{2}$$

Use the distributive property to multiply x+5 by 4.

$$8x^{2}+16x=4x+20-\left(x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$8x^{2}+16x=4x+20-\left(x^{2}+10x+25\right)$$

To find the opposite of x^{2}+10x+25, find the opposite of each term.

$$8x^{2}+16x=4x+20-x^{2}-10x-25$$

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

$$8x^{2}+16x=-6x+20-x^{2}-25$$

Subtract 25 from 20 to get -5.

$$8x^{2}+16x=-6x-5-x^{2}$$

Add 6x to both sides.

$$8x^{2}+16x+6x=-5-x^{2}$$

Combine 16x and 6x to get 22x.

$$8x^{2}+22x=-5-x^{2}$$

Subtract -5 from both sides.

$$8x^{2}+22x-\left(-5\right)=-x^{2}$$

The opposite of -5 is 5.

$$8x^{2}+22x+5=-x^{2}$$

Add x^{2} to both sides.

$$8x^{2}+22x+5+x^{2}=0$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$9x^{2}+22x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 22 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\times 9\times 5}}{2\times 9}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 9\times 5}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-22±\sqrt{484-36\times 5}}{2\times 9}$$

Multiply -36 times 5.

$$x=\frac{-22±\sqrt{484-180}}{2\times 9}$$

Add 484 to -180.

$$x=\frac{-22±\sqrt{304}}{2\times 9}$$

Take the square root of 304.

$$x=\frac{-22±4\sqrt{19}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-22±4\sqrt{19}}{18}$$

Now solve the equation x=\frac{-22±4\sqrt{19}}{18} when ± is plus. Add -22 to 4\sqrt{19}.

$$x=\frac{4\sqrt{19}-22}{18}$$

Divide -22+4\sqrt{19} by 18.

$$x=\frac{2\sqrt{19}-11}{9}$$

Now solve the equation x=\frac{-22±4\sqrt{19}}{18} when ± is minus. Subtract 4\sqrt{19} from -22.

$$x=\frac{-4\sqrt{19}-22}{18}$$

Divide -22-4\sqrt{19} by 18.

$$x=\frac{-2\sqrt{19}-11}{9}$$

The equation is now solved.

$$x=\frac{2\sqrt{19}-11}{9} x=\frac{-2\sqrt{19}-11}{9}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+5\right)^{2}, the least common multiple of x^{2}+10x+25,x^{2}+7x+10,x+2.

$$\left(x+2\right)\times 8x=\left(x+5\right)\times 4-\left(x+5\right)^{2}$$

Use the distributive property to multiply x+2 by 8.

$$\left(8x+16\right)x=\left(x+5\right)\times 4-\left(x+5\right)^{2}$$

Use the distributive property to multiply 8x+16 by x.

$$8x^{2}+16x=\left(x+5\right)\times 4-\left(x+5\right)^{2}$$

Use the distributive property to multiply x+5 by 4.

$$8x^{2}+16x=4x+20-\left(x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$8x^{2}+16x=4x+20-\left(x^{2}+10x+25\right)$$

To find the opposite of x^{2}+10x+25, find the opposite of each term.

$$8x^{2}+16x=4x+20-x^{2}-10x-25$$

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

$$8x^{2}+16x=-6x+20-x^{2}-25$$

Subtract 25 from 20 to get -5.

$$8x^{2}+16x=-6x-5-x^{2}$$

Add 6x to both sides.

$$8x^{2}+16x+6x=-5-x^{2}$$

Combine 16x and 6x to get 22x.

$$8x^{2}+22x=-5-x^{2}$$

Add x^{2} to both sides.

$$8x^{2}+22x+x^{2}=-5$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$9x^{2}+22x=-5$$

Divide both sides by 9.

$$\frac{9x^{2}+22x}{9}=-\frac{5}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{22}{9}x=-\frac{5}{9}$$

Divide \frac{22}{9}, the coefficient of the x term, by 2 to get \frac{11}{9}. Then add the square of \frac{11}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{22}{9}x+\left(\frac{11}{9}\right)^{2}=-\frac{5}{9}+\left(\frac{11}{9}\right)^{2}$$

Square \frac{11}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{22}{9}x+\frac{121}{81}=-\frac{5}{9}+\frac{121}{81}$$

Add -\frac{5}{9} to \frac{121}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{22}{9}x+\frac{121}{81}=\frac{76}{81}$$

Factor x^{2}+\frac{22}{9}x+\frac{121}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{9}\right)^{2}=\frac{76}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{9}\right)^{2}}=\sqrt{\frac{76}{81}}$$

Simplify.

$$x+\frac{11}{9}=\frac{2\sqrt{19}}{9} x+\frac{11}{9}=-\frac{2\sqrt{19}}{9}$$

Subtract \frac{11}{9} from both sides of the equation.

$$x=\frac{2\sqrt{19}-11}{9} x=\frac{-2\sqrt{19}-11}{9}$$