Solve for x
x=-65
x=50
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\left(x+15\right)\times 6500-x\times 6500=30x\left(x+15\right)
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 x\left(x+15\right), the least common multiple of x,x+15.
6500x+97500-x\times 6500=30x\left(x+15\right)
Use the distributive property to multiply x+15 by 6500.
6500x+97500-x\times 6500=30x^{2}+450x
Use the distributive property to multiply 30x by x+15.
6500x+97500-x\times 6500-30x^{2}=450x
Subtract 30x^{2} from both sides.
6500x+97500-x\times 6500-30x^{2}-450x=0
Subtract 450x from both sides.
6050x+97500-x\times 6500-30x^{2}=0
Combine 6500x and -450x to get 6050x.
6050x+97500-6500x-30x^{2}=0
Multiply -1 and 6500 to get -6500.
-450x+97500-30x^{2}=0
Combine 6050x and -6500x to get -450x.
-15x+3250-x^{2}=0
Divide both sides by 30.
-x^{2}-15x+3250=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-15 ab=-3250=-3250
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+3250. To find a and b, set up a system to be solved.
1,-3250 2,-1625 5,-650 10,-325 13,-250 25,-130 26,-125 50,-65
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 -3250.
1-3250=-3249 2-1625=-1623 5-650=-645 10-325=-315 13-250=-237 25-130=-105 26-125=-99 50-65=-15
Calculate the sum for each pair.
a=50 b=-65
The solution is the pair that gives sum -15.
\left(-x^{2}+50x\right)+\left(-65x+3250\right)
Rewrite -x^{2}-15x+3250 as \left(-x^{2}+50x\right)+\left(-65x+3250\right).
x\left(-x+50\right)+65\left(-x+50\right)
Factor out x in the first and 65 in the second group.
\left(-x+50\right)\left(x+65\right)
Factor out common term -x+50 by using distributive property.
x=50 x=-65
To find equation solutions, solve -x+50=0 and x+65=0.
\left(x+15\right)\times 6500-x\times 6500=30x\left(x+15\right)
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 x\left(x+15\right), the least common multiple of x,x+15.
6500x+97500-x\times 6500=30x\left(x+15\right)
Use the distributive property to multiply x+15 by 6500.
6500x+97500-x\times 6500=30x^{2}+450x
Use the distributive property to multiply 30x by x+15.
6500x+97500-x\times 6500-30x^{2}=450x
Subtract 30x^{2} from both sides.
6500x+97500-x\times 6500-30x^{2}-450x=0
Subtract 450x from both sides.
6050x+97500-x\times 6500-30x^{2}=0
Combine 6500x and -450x to get 6050x.
6050x+97500-6500x-30x^{2}=0
Multiply -1 and 6500 to get -6500.
-450x+97500-30x^{2}=0
Combine 6050x and -6500x to get -450x.
-30x^{2}-450x+97500=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(-450\right)±\sqrt{\left(-450\right)^{2}-4\left(-30\right)\times 97500}}{2\left(-30\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -30 for a, -450 for b, and 97500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-450\right)±\sqrt{202500-4\left(-30\right)\times 97500}}{2\left(-30\right)}
Square -450.
x=\frac{-\left(-450\right)±\sqrt{202500+120\times 97500}}{2\left(-30\right)}
Multiply -4 times -30.
x=\frac{-\left(-450\right)±\sqrt{202500+11700000}}{2\left(-30\right)}
Multiply 120 times 97500.
x=\frac{-\left(-450\right)±\sqrt{11902500}}{2\left(-30\right)}
Add 202500 to 11700000.
x=\frac{-\left(-450\right)±3450}{2\left(-30\right)}
Take the square root of 11902500.
x=\frac{450±3450}{2\left(-30\right)}
The opposite of -450 is 450.
x=\frac{450±3450}{-60}
Multiply 2 times -30.
x=\frac{3900}{-60}
Now solve the equation x=\frac{450±3450}{-60} when ± is plus. Add 450 to 3450.
x=-65
Divide 3900 by -60.
x=-\frac{3000}{-60}
Now solve the equation x=\frac{450±3450}{-60} when ± is minus. Subtract 3450 from 450.
x=50
Divide -3000 by -60.
x=-65 x=50
The equation is now solved.
\left(x+15\right)\times 6500-x\times 6500=30x\left(x+15\right)
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 x\left(x+15\right), the least common multiple of x,x+15.
6500x+97500-x\times 6500=30x\left(x+15\right)
Use the distributive property to multiply x+15 by 6500.
6500x+97500-x\times 6500=30x^{2}+450x
Use the distributive property to multiply 30x by x+15.
6500x+97500-x\times 6500-30x^{2}=450x
Subtract 30x^{2} from both sides.
6500x+97500-x\times 6500-30x^{2}-450x=0
Subtract 450x from both sides.
6050x+97500-x\times 6500-30x^{2}=0
Combine 6500x and -450x to get 6050x.
6050x-x\times 6500-30x^{2}=-97500
Subtract 97500 from both sides. Anything subtracted from zero gives its negation.
6050x-6500x-30x^{2}=-97500
Multiply -1 and 6500 to get -6500.
-450x-30x^{2}=-97500
Combine 6050x and -6500x to get -450x.
-30x^{2}-450x=-97500
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{-30x^{2}-450x}{-30}=-\frac{97500}{-30}
Divide both sides by -30.
x^{2}+\left(-\frac{450}{-30}\right)x=-\frac{97500}{-30}
Dividing by -30 undoes the multiplication by -30.
x^{2}+15x=-\frac{97500}{-30}
Divide -450 by -30.
x^{2}+15x=3250
Divide -97500 by -30.
x^{2}+15x+\left(\frac{15}{2}\right)^{2}=3250+\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}=3250+\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{13225}{4}
Add 3250 to \frac{225}{4}.
\left(x+\frac{15}{2}\right)^{2}=\frac{13225}{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{13225}{4}}
Take the square root of both sides of the equation.
x+\frac{15}{2}=\frac{115}{2} x+\frac{15}{2}=-\frac{115}{2}
Simplify.
x=50 x=-65
Subtract \frac{15}{2} from both sides of the equation.
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