Solve 3/2+262.5+3/2/frac{x+25{2}}=300/x+frac{1}{2}

$$\frac{ 3 }{ 2 } + \frac{ 262.5+ \frac{ 3 }{ 2 } }{ \frac{ x+25 }{ 2 } } = \frac{ 300 }{ x } + \frac{ 1 }{ 2 }$$

Solve for x

$$x = \frac{\sqrt{94009} - 253}{2} \approx 26.804435683$$

$$x=\frac{-\sqrt{94009}-253}{2}\approx -279.804435683$$

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.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(262.5+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

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

$$3x+4x\left(x+25\right)^{-1}\left(262.5+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Add 262.5 and \frac{3}{2} to get 264.

$$3x+4x\left(x+25\right)^{-1}\times 264=2\times 300+2x\times \frac{1}{2}$$

Multiply 4 and 264 to get 1056.

$$3x+1056x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and 300 to get 600.

$$3x+1056x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

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

$$3x+1056x\left(x+25\right)^{-1}=600+x$$

Subtract 600 from both sides.

$$3x+1056x\left(x+25\right)^{-1}-600=x$$

Subtract x from both sides.

$$3x+1056x\left(x+25\right)^{-1}-600-x=0$$

Combine 3x and -x to get 2x.

$$2x+1056x\left(x+25\right)^{-1}-600=0$$

Reorder the terms.

$$2x+1056\times \frac{1}{x+25}x-600=0$$

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

$$2x\left(x+25\right)+1056\times 1x+\left(x+25\right)\left(-600\right)=0$$

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

$$2x^{2}+50x+1056\times 1x+\left(x+25\right)\left(-600\right)=0$$

Multiply 1056 and 1 to get 1056.

$$2x^{2}+50x+1056x+\left(x+25\right)\left(-600\right)=0$$

Combine 50x and 1056x to get 1106x.

$$2x^{2}+1106x+\left(x+25\right)\left(-600\right)=0$$

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

$$2x^{2}+1106x-600x-15000=0$$

Combine 1106x and -600x to get 506x.

$$2x^{2}+506x-15000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 506 for b, and -15000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-506±\sqrt{506^{2}-4\times 2\left(-15000\right)}}{2\times 2}$$

Square 506.

$$x=\frac{-506±\sqrt{256036-4\times 2\left(-15000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-506±\sqrt{256036-8\left(-15000\right)}}{2\times 2}$$

Multiply -8 times -15000.

$$x=\frac{-506±\sqrt{256036+120000}}{2\times 2}$$

Add 256036 to 120000.

$$x=\frac{-506±\sqrt{376036}}{2\times 2}$$

Take the square root of 376036.

$$x=\frac{-506±2\sqrt{94009}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-506±2\sqrt{94009}}{4}$$

Now solve the equation x=\frac{-506±2\sqrt{94009}}{4} when ± is plus. Add -506 to 2\sqrt{94009}.

$$x=\frac{2\sqrt{94009}-506}{4}$$

Divide -506+2\sqrt{94009} by 4.

$$x=\frac{\sqrt{94009}-253}{2}$$

Now solve the equation x=\frac{-506±2\sqrt{94009}}{4} when ± is minus. Subtract 2\sqrt{94009} from -506.

$$x=\frac{-2\sqrt{94009}-506}{4}$$

Divide -506-2\sqrt{94009} by 4.

$$x=\frac{-\sqrt{94009}-253}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{94009}-253}{2} x=\frac{-\sqrt{94009}-253}{2}$$

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.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(262.5+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

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

$$3x+4x\left(x+25\right)^{-1}\left(262.5+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Add 262.5 and \frac{3}{2} to get 264.

$$3x+4x\left(x+25\right)^{-1}\times 264=2\times 300+2x\times \frac{1}{2}$$

Multiply 4 and 264 to get 1056.

$$3x+1056x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Multiply 2 and 300 to get 600.

$$3x+1056x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

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

$$3x+1056x\left(x+25\right)^{-1}=600+x$$

Subtract x from both sides.

$$3x+1056x\left(x+25\right)^{-1}-x=600$$

Combine 3x and -x to get 2x.

$$2x+1056x\left(x+25\right)^{-1}=600$$

Reorder the terms.

$$2x+1056\times \frac{1}{x+25}x=600$$

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

$$2x\left(x+25\right)+1056\times 1x=600\left(x+25\right)$$

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

$$2x^{2}+50x+1056\times 1x=600\left(x+25\right)$$

Multiply 1056 and 1 to get 1056.

$$2x^{2}+50x+1056x=600\left(x+25\right)$$

Combine 50x and 1056x to get 1106x.

$$2x^{2}+1106x=600\left(x+25\right)$$

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

$$2x^{2}+1106x=600x+15000$$

Subtract 600x from both sides.

$$2x^{2}+1106x-600x=15000$$

Combine 1106x and -600x to get 506x.

$$2x^{2}+506x=15000$$

Divide both sides by 2.

$$\frac{2x^{2}+506x}{2}=\frac{15000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{506}{2}x=\frac{15000}{2}$$

Divide 506 by 2.

$$x^{2}+253x=\frac{15000}{2}$$

Divide 15000 by 2.

$$x^{2}+253x=7500$$

Divide 253, the coefficient of the x term, by 2 to get \frac{253}{2}. Then add the square of \frac{253}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+253x+\left(\frac{253}{2}\right)^{2}=7500+\left(\frac{253}{2}\right)^{2}$$

Square \frac{253}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+253x+\frac{64009}{4}=7500+\frac{64009}{4}$$

Add 7500 to \frac{64009}{4}.

$$x^{2}+253x+\frac{64009}{4}=\frac{94009}{4}$$

Factor x^{2}+253x+\frac{64009}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{253}{2}\right)^{2}=\frac{94009}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{253}{2}\right)^{2}}=\sqrt{\frac{94009}{4}}$$

Simplify.

$$x+\frac{253}{2}=\frac{\sqrt{94009}}{2} x+\frac{253}{2}=-\frac{\sqrt{94009}}{2}$$

Subtract \frac{253}{2} from both sides of the equation.

$$x=\frac{\sqrt{94009}-253}{2} x=\frac{-\sqrt{94009}-253}{2}$$