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150x-x^{2}=\left(1-0\right)\times 100\times 50
Multiply 0 and 8832 to get 0.
150x-x^{2}=1\times 100\times 50
Subtract 0 from 1 to get 1.
150x-x^{2}=100\times 50
Multiply 1 and 100 to get 100.
150x-x^{2}=5000
Multiply 100 and 50 to get 5000.
150x-x^{2}-5000=0
Subtract 5000 from both sides.
-x^{2}+150x-5000=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{-150±\sqrt{150^{2}-4\left(-1\right)\left(-5000\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 150 for b, and -5000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-150±\sqrt{22500-4\left(-1\right)\left(-5000\right)}}{2\left(-1\right)}
Square 150.
x=\frac{-150±\sqrt{22500+4\left(-5000\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-150±\sqrt{22500-20000}}{2\left(-1\right)}
Multiply 4 times -5000.
x=\frac{-150±\sqrt{2500}}{2\left(-1\right)}
Add 22500 to -20000.
x=\frac{-150±50}{2\left(-1\right)}
Take the square root of 2500.
x=\frac{-150±50}{-2}
Multiply 2 times -1.
x=-\frac{100}{-2}
Now solve the equation x=\frac{-150±50}{-2} when ± is plus. Add -150 to 50.
x=50
Divide -100 by -2.
x=-\frac{200}{-2}
Now solve the equation x=\frac{-150±50}{-2} when ± is minus. Subtract 50 from -150.
x=100
Divide -200 by -2.
x=50 x=100
The equation is now solved.
150x-x^{2}=\left(1-0\right)\times 100\times 50
Multiply 0 and 8832 to get 0.
150x-x^{2}=1\times 100\times 50
Subtract 0 from 1 to get 1.
150x-x^{2}=100\times 50
Multiply 1 and 100 to get 100.
150x-x^{2}=5000
Multiply 100 and 50 to get 5000.
-x^{2}+150x=5000
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}+150x}{-1}=\frac{5000}{-1}
Divide both sides by -1.
x^{2}+\frac{150}{-1}x=\frac{5000}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-150x=\frac{5000}{-1}
Divide 150 by -1.
x^{2}-150x=-5000
Divide 5000 by -1.
x^{2}-150x+\left(-75\right)^{2}=-5000+\left(-75\right)^{2}
Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-150x+5625=-5000+5625
Square -75.
x^{2}-150x+5625=625
Add -5000 to 5625.
\left(x-75\right)^{2}=625
Factor x^{2}-150x+5625. 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-75\right)^{2}}=\sqrt{625}
Take the square root of both sides of the equation.
x-75=25 x-75=-25
Simplify.
x=100 x=50
Add 75 to both sides of the equation.