Solve 2x-1/2left(2x+3right)left(2x+5right)+2/2x+3=x+4/2xleft(x+2right)+5left(x+2right)

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

Solve for x (complex solution)

$$x=\frac{-17+\sqrt{47}i}{12}\approx -1.416666667+0.57130455i$$

$$x=\frac{-\sqrt{47}i-17}{12}\approx -1.416666667-0.57130455i$$

Steps Using the Quadratic Formula

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

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

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

$$3x-2+2x^{2}+2\left(2+x\right)\left(2x+5\right)\times 2=\left(4x+6\right)\left(x+4\right)$$

Multiply 2 and 2 to get 4.

$$3x-2+2x^{2}+4\left(2+x\right)\left(2x+5\right)=\left(4x+6\right)\left(x+4\right)$$

Use the distributive property to multiply 4 by 2+x.

$$3x-2+2x^{2}+\left(8+4x\right)\left(2x+5\right)=\left(4x+6\right)\left(x+4\right)$$

Use the distributive property to multiply 8+4x by 2x+5 and combine like terms.

$$3x-2+2x^{2}+36x+40+8x^{2}=\left(4x+6\right)\left(x+4\right)$$

Combine 3x and 36x to get 39x.

$$39x-2+2x^{2}+40+8x^{2}=\left(4x+6\right)\left(x+4\right)$$

Add -2 and 40 to get 38.

$$39x+38+2x^{2}+8x^{2}=\left(4x+6\right)\left(x+4\right)$$

Combine 2x^{2} and 8x^{2} to get 10x^{2}.

$$39x+38+10x^{2}=\left(4x+6\right)\left(x+4\right)$$

Use the distributive property to multiply 4x+6 by x+4 and combine like terms.

$$39x+38+10x^{2}=4x^{2}+22x+24$$

Subtract 4x^{2} from both sides.

$$39x+38+10x^{2}-4x^{2}=22x+24$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

$$39x+38+6x^{2}=22x+24$$

Subtract 22x from both sides.

$$39x+38+6x^{2}-22x=24$$

Combine 39x and -22x to get 17x.

$$17x+38+6x^{2}=24$$

Subtract 24 from both sides.

$$17x+38+6x^{2}-24=0$$

Subtract 24 from 38 to get 14.

$$17x+14+6x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}+17x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 17 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 6\times 14}}{2\times 6}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 6\times 14}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-17±\sqrt{289-24\times 14}}{2\times 6}$$

Multiply -24 times 14.

$$x=\frac{-17±\sqrt{289-336}}{2\times 6}$$

Add 289 to -336.

$$x=\frac{-17±\sqrt{-47}}{2\times 6}$$

Take the square root of -47.

$$x=\frac{-17±\sqrt{47}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-17±\sqrt{47}i}{12}$$

Now solve the equation x=\frac{-17±\sqrt{47}i}{12} when ± is plus. Add -17 to i\sqrt{47}.

$$x=\frac{-17+\sqrt{47}i}{12}$$

Now solve the equation x=\frac{-17±\sqrt{47}i}{12} when ± is minus. Subtract i\sqrt{47} from -17.

$$x=\frac{-\sqrt{47}i-17}{12}$$

The equation is now solved.

$$x=\frac{-17+\sqrt{47}i}{12} x=\frac{-\sqrt{47}i-17}{12}$$

Steps for Completing the Square

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

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

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

$$3x-2+2x^{2}+2\left(2+x\right)\left(2x+5\right)\times 2=\left(4x+6\right)\left(x+4\right)$$

Multiply 2 and 2 to get 4.

$$3x-2+2x^{2}+4\left(2+x\right)\left(2x+5\right)=\left(4x+6\right)\left(x+4\right)$$

Use the distributive property to multiply 4 by 2+x.

$$3x-2+2x^{2}+\left(8+4x\right)\left(2x+5\right)=\left(4x+6\right)\left(x+4\right)$$

Use the distributive property to multiply 8+4x by 2x+5 and combine like terms.

$$3x-2+2x^{2}+36x+40+8x^{2}=\left(4x+6\right)\left(x+4\right)$$

Combine 3x and 36x to get 39x.

$$39x-2+2x^{2}+40+8x^{2}=\left(4x+6\right)\left(x+4\right)$$

Add -2 and 40 to get 38.

$$39x+38+2x^{2}+8x^{2}=\left(4x+6\right)\left(x+4\right)$$

Combine 2x^{2} and 8x^{2} to get 10x^{2}.

$$39x+38+10x^{2}=\left(4x+6\right)\left(x+4\right)$$

Use the distributive property to multiply 4x+6 by x+4 and combine like terms.

$$39x+38+10x^{2}=4x^{2}+22x+24$$

Subtract 4x^{2} from both sides.

$$39x+38+10x^{2}-4x^{2}=22x+24$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

$$39x+38+6x^{2}=22x+24$$

Subtract 22x from both sides.

$$39x+38+6x^{2}-22x=24$$

Combine 39x and -22x to get 17x.

$$17x+38+6x^{2}=24$$

Subtract 38 from both sides.

$$17x+6x^{2}=24-38$$

Subtract 38 from 24 to get -14.

$$17x+6x^{2}=-14$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}+17x=-14$$

Divide both sides by 6.

$$\frac{6x^{2}+17x}{6}=-\frac{14}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{17}{6}x=-\frac{14}{6}$$

Reduce the fraction \frac{-14}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{17}{6}x=-\frac{7}{3}$$

Divide \frac{17}{6}, the coefficient of the x term, by 2 to get \frac{17}{12}. Then add the square of \frac{17}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{17}{6}x+\left(\frac{17}{12}\right)^{2}=-\frac{7}{3}+\left(\frac{17}{12}\right)^{2}$$

Square \frac{17}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{17}{6}x+\frac{289}{144}=-\frac{7}{3}+\frac{289}{144}$$

Add -\frac{7}{3} to \frac{289}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{17}{6}x+\frac{289}{144}=-\frac{47}{144}$$

Factor x^{2}+\frac{17}{6}x+\frac{289}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{17}{12}\right)^{2}=-\frac{47}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{12}\right)^{2}}=\sqrt{-\frac{47}{144}}$$

Simplify.

$$x+\frac{17}{12}=\frac{\sqrt{47}i}{12} x+\frac{17}{12}=-\frac{\sqrt{47}i}{12}$$

Subtract \frac{17}{12} from both sides of the equation.

$$x=\frac{-17+\sqrt{47}i}{12} x=\frac{-\sqrt{47}i-17}{12}$$