Solve 3/2+2625+3/2/frac{x+25{2}}=300/x+frac{1}{2}

$$\frac{ 3 }{ 2 } + \frac{ 2625+ \frac{ 3 }{ 2 } }{ \frac{ x+25 }{ 2 } } = \frac{ 300 }{ x } + \frac{ 1 }{ 2 }$$

Solve for x (complex solution)

$$x=\sqrt{6202621}-2489\approx 1.506173451$$

$$x=-\left(\sqrt{6202621}+2489\right)\approx -4979.506173451$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and \frac{3}{2} to get 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Add 2625 and \frac{3}{2} to get \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Multiply 4 and \frac{5253}{2} to get 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and 300 to get 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Subtract 600 from both sides.

$$3x+10506x\left(x+25\right)^{-1}-600=x$$

Subtract x from both sides.

$$3x+10506x\left(x+25\right)^{-1}-600-x=0$$

Combine 3x and -x to get 2x.

$$2x+10506x\left(x+25\right)^{-1}-600=0$$

Reorder the terms.

$$2x+10506\times \frac{1}{x+25}x-600=0$$

Variable x cannot be equal to -25 since division by zero is not defined. Multiply both sides of the equation by x+25.

$$2x\left(x+25\right)+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Use the distributive property to multiply 2x by x+25.

$$2x^{2}+50x+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Multiply 10506 and 1 to get 10506.

$$2x^{2}+50x+10506x+\left(x+25\right)\left(-600\right)=0$$

Combine 50x and 10506x to get 10556x.

$$2x^{2}+10556x+\left(x+25\right)\left(-600\right)=0$$

Use the distributive property to multiply x+25 by -600.

$$2x^{2}+10556x-600x-15000=0$$

Combine 10556x and -600x to get 9956x.

$$2x^{2}+9956x-15000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 9956 for b, and -15000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9956±\sqrt{9956^{2}-4\times 2\left(-15000\right)}}{2\times 2}$$

Square 9956.

$$x=\frac{-9956±\sqrt{99121936-4\times 2\left(-15000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-9956±\sqrt{99121936-8\left(-15000\right)}}{2\times 2}$$

Multiply -8 times -15000.

$$x=\frac{-9956±\sqrt{99121936+120000}}{2\times 2}$$

Add 99121936 to 120000.

$$x=\frac{-9956±\sqrt{99241936}}{2\times 2}$$

Take the square root of 99241936.

$$x=\frac{-9956±4\sqrt{6202621}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-9956±4\sqrt{6202621}}{4}$$

Now solve the equation x=\frac{-9956±4\sqrt{6202621}}{4} when ± is plus. Add -9956 to 4\sqrt{6202621}.

$$x=\frac{4\sqrt{6202621}-9956}{4}$$

Divide -9956+4\sqrt{6202621} by 4.

$$x=\sqrt{6202621}-2489$$

Now solve the equation x=\frac{-9956±4\sqrt{6202621}}{4} when ± is minus. Subtract 4\sqrt{6202621} from -9956.

$$x=\frac{-4\sqrt{6202621}-9956}{4}$$

Divide -9956-4\sqrt{6202621} by 4.

$$x=-\sqrt{6202621}-2489$$

The equation is now solved.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and \frac{3}{2} to get 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Add 2625 and \frac{3}{2} to get \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Multiply 4 and \frac{5253}{2} to get 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and 300 to get 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Subtract x from both sides.

$$3x+10506x\left(x+25\right)^{-1}-x=600$$

Combine 3x and -x to get 2x.

$$2x+10506x\left(x+25\right)^{-1}=600$$

Reorder the terms.

$$2x+10506\times \frac{1}{x+25}x=600$$

Variable x cannot be equal to -25 since division by zero is not defined. Multiply both sides of the equation by x+25.

$$2x\left(x+25\right)+10506\times 1x=600\left(x+25\right)$$

Use the distributive property to multiply 2x by x+25.

$$2x^{2}+50x+10506\times 1x=600\left(x+25\right)$$

Multiply 10506 and 1 to get 10506.

$$2x^{2}+50x+10506x=600\left(x+25\right)$$

Combine 50x and 10506x to get 10556x.

$$2x^{2}+10556x=600\left(x+25\right)$$

Use the distributive property to multiply 600 by x+25.

$$2x^{2}+10556x=600x+15000$$

Subtract 600x from both sides.

$$2x^{2}+10556x-600x=15000$$

Combine 10556x and -600x to get 9956x.

$$2x^{2}+9956x=15000$$

Divide both sides by 2.

$$\frac{2x^{2}+9956x}{2}=\frac{15000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{9956}{2}x=\frac{15000}{2}$$

Divide 9956 by 2.

$$x^{2}+4978x=\frac{15000}{2}$$

Divide 15000 by 2.

$$x^{2}+4978x=7500$$

Divide 4978, the coefficient of the x term, by 2 to get 2489. Then add the square of 2489 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4978x+2489^{2}=7500+2489^{2}$$

Square 2489.

$$x^{2}+4978x+6195121=7500+6195121$$

Add 7500 to 6195121.

$$x^{2}+4978x+6195121=6202621$$

Factor x^{2}+4978x+6195121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2489\right)^{2}=6202621$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2489\right)^{2}}=\sqrt{6202621}$$

Simplify.

$$x+2489=\sqrt{6202621} x+2489=-\sqrt{6202621}$$

Subtract 2489 from both sides of the equation.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$

Solve for x

$$x=\sqrt{6202621}-2489\approx 1.506173451$$

$$x=-\sqrt{6202621}-2489\approx -4979.506173451$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and \frac{3}{2} to get 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Add 2625 and \frac{3}{2} to get \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Multiply 4 and \frac{5253}{2} to get 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and 300 to get 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Subtract 600 from both sides.

$$3x+10506x\left(x+25\right)^{-1}-600=x$$

Subtract x from both sides.

$$3x+10506x\left(x+25\right)^{-1}-600-x=0$$

Combine 3x and -x to get 2x.

$$2x+10506x\left(x+25\right)^{-1}-600=0$$

Reorder the terms.

$$2x+10506\times \frac{1}{x+25}x-600=0$$

Variable x cannot be equal to -25 since division by zero is not defined. Multiply both sides of the equation by x+25.

$$2x\left(x+25\right)+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Use the distributive property to multiply 2x by x+25.

$$2x^{2}+50x+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Multiply 10506 and 1 to get 10506.

$$2x^{2}+50x+10506x+\left(x+25\right)\left(-600\right)=0$$

Combine 50x and 10506x to get 10556x.

$$2x^{2}+10556x+\left(x+25\right)\left(-600\right)=0$$

Use the distributive property to multiply x+25 by -600.

$$2x^{2}+10556x-600x-15000=0$$

Combine 10556x and -600x to get 9956x.

$$2x^{2}+9956x-15000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 9956 for b, and -15000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9956±\sqrt{9956^{2}-4\times 2\left(-15000\right)}}{2\times 2}$$

Square 9956.

$$x=\frac{-9956±\sqrt{99121936-4\times 2\left(-15000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-9956±\sqrt{99121936-8\left(-15000\right)}}{2\times 2}$$

Multiply -8 times -15000.

$$x=\frac{-9956±\sqrt{99121936+120000}}{2\times 2}$$

Add 99121936 to 120000.

$$x=\frac{-9956±\sqrt{99241936}}{2\times 2}$$

Take the square root of 99241936.

$$x=\frac{-9956±4\sqrt{6202621}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-9956±4\sqrt{6202621}}{4}$$

Now solve the equation x=\frac{-9956±4\sqrt{6202621}}{4} when ± is plus. Add -9956 to 4\sqrt{6202621}.

$$x=\frac{4\sqrt{6202621}-9956}{4}$$

Divide -9956+4\sqrt{6202621} by 4.

$$x=\sqrt{6202621}-2489$$

Now solve the equation x=\frac{-9956±4\sqrt{6202621}}{4} when ± is minus. Subtract 4\sqrt{6202621} from -9956.

$$x=\frac{-4\sqrt{6202621}-9956}{4}$$

Divide -9956-4\sqrt{6202621} by 4.

$$x=-\sqrt{6202621}-2489$$

The equation is now solved.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and \frac{3}{2} to get 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Add 2625 and \frac{3}{2} to get \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Multiply 4 and \frac{5253}{2} to get 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and 300 to get 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Subtract x from both sides.

$$3x+10506x\left(x+25\right)^{-1}-x=600$$

Combine 3x and -x to get 2x.

$$2x+10506x\left(x+25\right)^{-1}=600$$

Reorder the terms.

$$2x+10506\times \frac{1}{x+25}x=600$$

Variable x cannot be equal to -25 since division by zero is not defined. Multiply both sides of the equation by x+25.

$$2x\left(x+25\right)+10506\times 1x=600\left(x+25\right)$$

Use the distributive property to multiply 2x by x+25.

$$2x^{2}+50x+10506\times 1x=600\left(x+25\right)$$

Multiply 10506 and 1 to get 10506.

$$2x^{2}+50x+10506x=600\left(x+25\right)$$

Combine 50x and 10506x to get 10556x.

$$2x^{2}+10556x=600\left(x+25\right)$$

Use the distributive property to multiply 600 by x+25.

$$2x^{2}+10556x=600x+15000$$

Subtract 600x from both sides.

$$2x^{2}+10556x-600x=15000$$

Combine 10556x and -600x to get 9956x.

$$2x^{2}+9956x=15000$$

Divide both sides by 2.

$$\frac{2x^{2}+9956x}{2}=\frac{15000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{9956}{2}x=\frac{15000}{2}$$

Divide 9956 by 2.

$$x^{2}+4978x=\frac{15000}{2}$$

Divide 15000 by 2.

$$x^{2}+4978x=7500$$

Divide 4978, the coefficient of the x term, by 2 to get 2489. Then add the square of 2489 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4978x+2489^{2}=7500+2489^{2}$$

Square 2489.

$$x^{2}+4978x+6195121=7500+6195121$$

Add 7500 to 6195121.

$$x^{2}+4978x+6195121=6202621$$

Factor x^{2}+4978x+6195121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2489\right)^{2}=6202621$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2489\right)^{2}}=\sqrt{6202621}$$

Simplify.

$$x+2489=\sqrt{6202621} x+2489=-\sqrt{6202621}$$

Subtract 2489 from both sides of the equation.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$