Solve for x
x = -\frac{20}{9} = -2\frac{2}{9} \approx -2.222222222
x=2
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Polynomial
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\frac{ x+2 }{ x+3 } - \frac{ x }{ x+5 } - \frac{ 18 }{ 35 } = 0
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\left(35x+175\right)\left(x+2\right)-\left(35x+105\right)x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 35\left(x+3\right)\left(x+5\right), the least common multiple of x+3,x+5,35.
35x^{2}+245x+350-\left(35x+105\right)x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Use the distributive property to multiply 35x+175 by x+2 and combine like terms.
35x^{2}+245x+350-\left(35x^{2}+105x\right)+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Use the distributive property to multiply 35x+105 by x.
35x^{2}+245x+350-35x^{2}-105x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
To find the opposite of 35x^{2}+105x, find the opposite of each term.
245x+350-105x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Combine 35x^{2} and -35x^{2} to get 0.
140x+350+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Combine 245x and -105x to get 140x.
140x+350-18\left(x+3\right)\left(x+5\right)=0
Multiply 35 and -\frac{18}{35} to get -18.
140x+350+\left(-18x-54\right)\left(x+5\right)=0
Use the distributive property to multiply -18 by x+3.
140x+350-18x^{2}-144x-270=0
Use the distributive property to multiply -18x-54 by x+5 and combine like terms.
-4x+350-18x^{2}-270=0
Combine 140x and -144x to get -4x.
-4x+80-18x^{2}=0
Subtract 270 from 350 to get 80.
-2x+40-9x^{2}=0
Divide both sides by 2.
-9x^{2}-2x+40=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-2 ab=-9\times 40=-360
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx+40. To find a and b, set up a system to be solved.
1,-360 2,-180 3,-120 4,-90 5,-72 6,-60 8,-45 9,-40 10,-36 12,-30 15,-24 18,-20
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 -360.
1-360=-359 2-180=-178 3-120=-117 4-90=-86 5-72=-67 6-60=-54 8-45=-37 9-40=-31 10-36=-26 12-30=-18 15-24=-9 18-20=-2
Calculate the sum for each pair.
a=18 b=-20
The solution is the pair that gives sum -2.
\left(-9x^{2}+18x\right)+\left(-20x+40\right)
Rewrite -9x^{2}-2x+40 as \left(-9x^{2}+18x\right)+\left(-20x+40\right).
9x\left(-x+2\right)+20\left(-x+2\right)
Factor out 9x in the first and 20 in the second group.
\left(-x+2\right)\left(9x+20\right)
Factor out common term -x+2 by using distributive property.
x=2 x=-\frac{20}{9}
To find equation solutions, solve -x+2=0 and 9x+20=0.
\left(35x+175\right)\left(x+2\right)-\left(35x+105\right)x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 35\left(x+3\right)\left(x+5\right), the least common multiple of x+3,x+5,35.
35x^{2}+245x+350-\left(35x+105\right)x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Use the distributive property to multiply 35x+175 by x+2 and combine like terms.
35x^{2}+245x+350-\left(35x^{2}+105x\right)+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Use the distributive property to multiply 35x+105 by x.
35x^{2}+245x+350-35x^{2}-105x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
To find the opposite of 35x^{2}+105x, find the opposite of each term.
245x+350-105x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Combine 35x^{2} and -35x^{2} to get 0.
140x+350+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Combine 245x and -105x to get 140x.
140x+350-18\left(x+3\right)\left(x+5\right)=0
Multiply 35 and -\frac{18}{35} to get -18.
140x+350+\left(-18x-54\right)\left(x+5\right)=0
Use the distributive property to multiply -18 by x+3.
140x+350-18x^{2}-144x-270=0
Use the distributive property to multiply -18x-54 by x+5 and combine like terms.
-4x+350-18x^{2}-270=0
Combine 140x and -144x to get -4x.
-4x+80-18x^{2}=0
Subtract 270 from 350 to get 80.
-18x^{2}-4x+80=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(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-18\right)\times 80}}{2\left(-18\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, -4 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\left(-18\right)\times 80}}{2\left(-18\right)}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16+72\times 80}}{2\left(-18\right)}
Multiply -4 times -18.
x=\frac{-\left(-4\right)±\sqrt{16+5760}}{2\left(-18\right)}
Multiply 72 times 80.
x=\frac{-\left(-4\right)±\sqrt{5776}}{2\left(-18\right)}
Add 16 to 5760.
x=\frac{-\left(-4\right)±76}{2\left(-18\right)}
Take the square root of 5776.
x=\frac{4±76}{2\left(-18\right)}
The opposite of -4 is 4.
x=\frac{4±76}{-36}
Multiply 2 times -18.
x=\frac{80}{-36}
Now solve the equation x=\frac{4±76}{-36} when ± is plus. Add 4 to 76.
x=-\frac{20}{9}
Reduce the fraction \frac{80}{-36} to lowest terms by extracting and canceling out 4.
x=-\frac{72}{-36}
Now solve the equation x=\frac{4±76}{-36} when ± is minus. Subtract 76 from 4.
x=2
Divide -72 by -36.
x=-\frac{20}{9} x=2
The equation is now solved.
\left(35x+175\right)\left(x+2\right)-\left(35x+105\right)x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 35\left(x+3\right)\left(x+5\right), the least common multiple of x+3,x+5,35.
35x^{2}+245x+350-\left(35x+105\right)x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Use the distributive property to multiply 35x+175 by x+2 and combine like terms.
35x^{2}+245x+350-\left(35x^{2}+105x\right)+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Use the distributive property to multiply 35x+105 by x.
35x^{2}+245x+350-35x^{2}-105x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
To find the opposite of 35x^{2}+105x, find the opposite of each term.
245x+350-105x+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Combine 35x^{2} and -35x^{2} to get 0.
140x+350+35\left(x+3\right)\left(x+5\right)\left(-\frac{18}{35}\right)=0
Combine 245x and -105x to get 140x.
140x+350-18\left(x+3\right)\left(x+5\right)=0
Multiply 35 and -\frac{18}{35} to get -18.
140x+350+\left(-18x-54\right)\left(x+5\right)=0
Use the distributive property to multiply -18 by x+3.
140x+350-18x^{2}-144x-270=0
Use the distributive property to multiply -18x-54 by x+5 and combine like terms.
-4x+350-18x^{2}-270=0
Combine 140x and -144x to get -4x.
-4x+80-18x^{2}=0
Subtract 270 from 350 to get 80.
-4x-18x^{2}=-80
Subtract 80 from both sides. Anything subtracted from zero gives its negation.
-18x^{2}-4x=-80
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{-18x^{2}-4x}{-18}=-\frac{80}{-18}
Divide both sides by -18.
x^{2}+\left(-\frac{4}{-18}\right)x=-\frac{80}{-18}
Dividing by -18 undoes the multiplication by -18.
x^{2}+\frac{2}{9}x=-\frac{80}{-18}
Reduce the fraction \frac{-4}{-18} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{2}{9}x=\frac{40}{9}
Reduce the fraction \frac{-80}{-18} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{2}{9}x+\left(\frac{1}{9}\right)^{2}=\frac{40}{9}+\left(\frac{1}{9}\right)^{2}
Divide \frac{2}{9}, the coefficient of the x term, by 2 to get \frac{1}{9}. Then add the square of \frac{1}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{2}{9}x+\frac{1}{81}=\frac{40}{9}+\frac{1}{81}
Square \frac{1}{9} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{2}{9}x+\frac{1}{81}=\frac{361}{81}
Add \frac{40}{9} to \frac{1}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{9}\right)^{2}=\frac{361}{81}
Factor x^{2}+\frac{2}{9}x+\frac{1}{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}.
\sqrt{\left(x+\frac{1}{9}\right)^{2}}=\sqrt{\frac{361}{81}}
Take the square root of both sides of the equation.
x+\frac{1}{9}=\frac{19}{9} x+\frac{1}{9}=-\frac{19}{9}
Simplify.
x=2 x=-\frac{20}{9}
Subtract \frac{1}{9} from both sides of the equation.
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{ x } ^ { 2 } - 4 x - 5 = 0
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Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}