Solve for x
x=-75
x=60
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Polynomial
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\frac { 75 } { x } = \frac { 75 } { x + 15 } + \frac { 1 } { 4 }
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\left(4x+60\right)\times 75=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}
Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+15\right), the least common multiple of x,x+15,4.
300x+4500=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}
Use the distributive property to multiply 4x+60 by 75.
300x+4500=300x+4x\left(x+15\right)\times \frac{1}{4}
Multiply 4 and 75 to get 300.
300x+4500=300x+x\left(x+15\right)
Multiply 4 and \frac{1}{4} to get 1.
300x+4500=300x+x^{2}+15x
Use the distributive property to multiply x by x+15.
300x+4500=315x+x^{2}
Combine 300x and 15x to get 315x.
300x+4500-315x=x^{2}
Subtract 315x from both sides.
-15x+4500=x^{2}
Combine 300x and -315x to get -15x.
-15x+4500-x^{2}=0
Subtract x^{2} from both sides.
-x^{2}-15x+4500=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-15 ab=-4500=-4500
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+4500. To find a and b, set up a system to be solved.
1,-4500 2,-2250 3,-1500 4,-1125 5,-900 6,-750 9,-500 10,-450 12,-375 15,-300 18,-250 20,-225 25,-180 30,-150 36,-125 45,-100 50,-90 60,-75
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4500.
1-4500=-4499 2-2250=-2248 3-1500=-1497 4-1125=-1121 5-900=-895 6-750=-744 9-500=-491 10-450=-440 12-375=-363 15-300=-285 18-250=-232 20-225=-205 25-180=-155 30-150=-120 36-125=-89 45-100=-55 50-90=-40 60-75=-15
Calculate the sum for each pair.
a=60 b=-75
The solution is the pair that gives sum -15.
\left(-x^{2}+60x\right)+\left(-75x+4500\right)
Rewrite -x^{2}-15x+4500 as \left(-x^{2}+60x\right)+\left(-75x+4500\right).
x\left(-x+60\right)+75\left(-x+60\right)
Factor out x in the first and 75 in the second group.
\left(-x+60\right)\left(x+75\right)
Factor out common term -x+60 by using distributive property.
x=60 x=-75
To find equation solutions, solve -x+60=0 and x+75=0.
\left(4x+60\right)\times 75=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}
Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+15\right), the least common multiple of x,x+15,4.
300x+4500=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}
Use the distributive property to multiply 4x+60 by 75.
300x+4500=300x+4x\left(x+15\right)\times \frac{1}{4}
Multiply 4 and 75 to get 300.
300x+4500=300x+x\left(x+15\right)
Multiply 4 and \frac{1}{4} to get 1.
300x+4500=300x+x^{2}+15x
Use the distributive property to multiply x by x+15.
300x+4500=315x+x^{2}
Combine 300x and 15x to get 315x.
300x+4500-315x=x^{2}
Subtract 315x from both sides.
-15x+4500=x^{2}
Combine 300x and -315x to get -15x.
-15x+4500-x^{2}=0
Subtract x^{2} from both sides.
-x^{2}-15x+4500=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.
x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-1\right)\times 4500}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -15 for b, and 4500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-15\right)±\sqrt{225-4\left(-1\right)\times 4500}}{2\left(-1\right)}
Square -15.
x=\frac{-\left(-15\right)±\sqrt{225+4\times 4500}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-15\right)±\sqrt{225+18000}}{2\left(-1\right)}
Multiply 4 times 4500.
x=\frac{-\left(-15\right)±\sqrt{18225}}{2\left(-1\right)}
Add 225 to 18000.
x=\frac{-\left(-15\right)±135}{2\left(-1\right)}
Take the square root of 18225.
x=\frac{15±135}{2\left(-1\right)}
The opposite of -15 is 15.
x=\frac{15±135}{-2}
Multiply 2 times -1.
x=\frac{150}{-2}
Now solve the equation x=\frac{15±135}{-2} when ± is plus. Add 15 to 135.
x=-75
Divide 150 by -2.
x=-\frac{120}{-2}
Now solve the equation x=\frac{15±135}{-2} when ± is minus. Subtract 135 from 15.
x=60
Divide -120 by -2.
x=-75 x=60
The equation is now solved.
\left(4x+60\right)\times 75=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}
Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+15\right), the least common multiple of x,x+15,4.
300x+4500=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}
Use the distributive property to multiply 4x+60 by 75.
300x+4500=300x+4x\left(x+15\right)\times \frac{1}{4}
Multiply 4 and 75 to get 300.
300x+4500=300x+x\left(x+15\right)
Multiply 4 and \frac{1}{4} to get 1.
300x+4500=300x+x^{2}+15x
Use the distributive property to multiply x by x+15.
300x+4500=315x+x^{2}
Combine 300x and 15x to get 315x.
300x+4500-315x=x^{2}
Subtract 315x from both sides.
-15x+4500=x^{2}
Combine 300x and -315x to get -15x.
-15x+4500-x^{2}=0
Subtract x^{2} from both sides.
-15x-x^{2}=-4500
Subtract 4500 from both sides. Anything subtracted from zero gives its negation.
-x^{2}-15x=-4500
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.
\frac{-x^{2}-15x}{-1}=-\frac{4500}{-1}
Divide both sides by -1.
x^{2}+\left(-\frac{15}{-1}\right)x=-\frac{4500}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}+15x=-\frac{4500}{-1}
Divide -15 by -1.
x^{2}+15x=4500
Divide -4500 by -1.
x^{2}+15x+\left(\frac{15}{2}\right)^{2}=4500+\left(\frac{15}{2}\right)^{2}
Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+15x+\frac{225}{4}=4500+\frac{225}{4}
Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+15x+\frac{225}{4}=\frac{18225}{4}
Add 4500 to \frac{225}{4}.
\left(x+\frac{15}{2}\right)^{2}=\frac{18225}{4}
Factor x^{2}+15x+\frac{225}{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}.
\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{18225}{4}}
Take the square root of both sides of the equation.
x+\frac{15}{2}=\frac{135}{2} x+\frac{15}{2}=-\frac{135}{2}
Simplify.
x=60 x=-75
Subtract \frac{15}{2} from both sides of the equation.
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