Solve for y
y=\frac{1}{60}\approx 0.016666667
y=\frac{1}{10}=0.1
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20y-1-\left(-y\times 20\right)=30y\left(20y-1\right)
Variable y cannot be equal to any of the values 0,\frac{1}{20} since division by zero is not defined. Multiply both sides of the equation by y\left(20y-1\right), the least common multiple of y,1-20y.
20y-1-\left(-20y\right)=30y\left(20y-1\right)
Multiply -1 and 20 to get -20.
20y-1+20y=30y\left(20y-1\right)
The opposite of -20y is 20y.
40y-1=30y\left(20y-1\right)
Combine 20y and 20y to get 40y.
40y-1=600y^{2}-30y
Use the distributive property to multiply 30y by 20y-1.
40y-1-600y^{2}=-30y
Subtract 600y^{2} from both sides.
40y-1-600y^{2}+30y=0
Add 30y to both sides.
70y-1-600y^{2}=0
Combine 40y and 30y to get 70y.
-600y^{2}+70y-1=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=70 ab=-600\left(-1\right)=600
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -600y^{2}+ay+by-1. To find a and b, set up a system to be solved.
1,600 2,300 3,200 4,150 5,120 6,100 8,75 10,60 12,50 15,40 20,30 24,25
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 600.
1+600=601 2+300=302 3+200=203 4+150=154 5+120=125 6+100=106 8+75=83 10+60=70 12+50=62 15+40=55 20+30=50 24+25=49
Calculate the sum for each pair.
a=60 b=10
The solution is the pair that gives sum 70.
\left(-600y^{2}+60y\right)+\left(10y-1\right)
Rewrite -600y^{2}+70y-1 as \left(-600y^{2}+60y\right)+\left(10y-1\right).
-60y\left(10y-1\right)+10y-1
Factor out -60y in -600y^{2}+60y.
\left(10y-1\right)\left(-60y+1\right)
Factor out common term 10y-1 by using distributive property.
y=\frac{1}{10} y=\frac{1}{60}
To find equation solutions, solve 10y-1=0 and -60y+1=0.
20y-1-\left(-y\times 20\right)=30y\left(20y-1\right)
Variable y cannot be equal to any of the values 0,\frac{1}{20} since division by zero is not defined. Multiply both sides of the equation by y\left(20y-1\right), the least common multiple of y,1-20y.
20y-1-\left(-20y\right)=30y\left(20y-1\right)
Multiply -1 and 20 to get -20.
20y-1+20y=30y\left(20y-1\right)
The opposite of -20y is 20y.
40y-1=30y\left(20y-1\right)
Combine 20y and 20y to get 40y.
40y-1=600y^{2}-30y
Use the distributive property to multiply 30y by 20y-1.
40y-1-600y^{2}=-30y
Subtract 600y^{2} from both sides.
40y-1-600y^{2}+30y=0
Add 30y to both sides.
70y-1-600y^{2}=0
Combine 40y and 30y to get 70y.
-600y^{2}+70y-1=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.
y=\frac{-70±\sqrt{70^{2}-4\left(-600\right)\left(-1\right)}}{2\left(-600\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -600 for a, 70 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-70±\sqrt{4900-4\left(-600\right)\left(-1\right)}}{2\left(-600\right)}
Square 70.
y=\frac{-70±\sqrt{4900+2400\left(-1\right)}}{2\left(-600\right)}
Multiply -4 times -600.
y=\frac{-70±\sqrt{4900-2400}}{2\left(-600\right)}
Multiply 2400 times -1.
y=\frac{-70±\sqrt{2500}}{2\left(-600\right)}
Add 4900 to -2400.
y=\frac{-70±50}{2\left(-600\right)}
Take the square root of 2500.
y=\frac{-70±50}{-1200}
Multiply 2 times -600.
y=-\frac{20}{-1200}
Now solve the equation y=\frac{-70±50}{-1200} when ± is plus. Add -70 to 50.
y=\frac{1}{60}
Reduce the fraction \frac{-20}{-1200} to lowest terms by extracting and canceling out 20.
y=-\frac{120}{-1200}
Now solve the equation y=\frac{-70±50}{-1200} when ± is minus. Subtract 50 from -70.
y=\frac{1}{10}
Reduce the fraction \frac{-120}{-1200} to lowest terms by extracting and canceling out 120.
y=\frac{1}{60} y=\frac{1}{10}
The equation is now solved.
20y-1-\left(-y\times 20\right)=30y\left(20y-1\right)
Variable y cannot be equal to any of the values 0,\frac{1}{20} since division by zero is not defined. Multiply both sides of the equation by y\left(20y-1\right), the least common multiple of y,1-20y.
20y-1-\left(-20y\right)=30y\left(20y-1\right)
Multiply -1 and 20 to get -20.
20y-1+20y=30y\left(20y-1\right)
The opposite of -20y is 20y.
40y-1=30y\left(20y-1\right)
Combine 20y and 20y to get 40y.
40y-1=600y^{2}-30y
Use the distributive property to multiply 30y by 20y-1.
40y-1-600y^{2}=-30y
Subtract 600y^{2} from both sides.
40y-1-600y^{2}+30y=0
Add 30y to both sides.
70y-1-600y^{2}=0
Combine 40y and 30y to get 70y.
70y-600y^{2}=1
Add 1 to both sides. Anything plus zero gives itself.
-600y^{2}+70y=1
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{-600y^{2}+70y}{-600}=\frac{1}{-600}
Divide both sides by -600.
y^{2}+\frac{70}{-600}y=\frac{1}{-600}
Dividing by -600 undoes the multiplication by -600.
y^{2}-\frac{7}{60}y=\frac{1}{-600}
Reduce the fraction \frac{70}{-600} to lowest terms by extracting and canceling out 10.
y^{2}-\frac{7}{60}y=-\frac{1}{600}
Divide 1 by -600.
y^{2}-\frac{7}{60}y+\left(-\frac{7}{120}\right)^{2}=-\frac{1}{600}+\left(-\frac{7}{120}\right)^{2}
Divide -\frac{7}{60}, the coefficient of the x term, by 2 to get -\frac{7}{120}. Then add the square of -\frac{7}{120} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}-\frac{7}{60}y+\frac{49}{14400}=-\frac{1}{600}+\frac{49}{14400}
Square -\frac{7}{120} by squaring both the numerator and the denominator of the fraction.
y^{2}-\frac{7}{60}y+\frac{49}{14400}=\frac{1}{576}
Add -\frac{1}{600} to \frac{49}{14400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(y-\frac{7}{120}\right)^{2}=\frac{1}{576}
Factor y^{2}-\frac{7}{60}y+\frac{49}{14400}. 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(y-\frac{7}{120}\right)^{2}}=\sqrt{\frac{1}{576}}
Take the square root of both sides of the equation.
y-\frac{7}{120}=\frac{1}{24} y-\frac{7}{120}=-\frac{1}{24}
Simplify.
y=\frac{1}{10} y=\frac{1}{60}
Add \frac{7}{120} to both sides of the equation.
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