Solve for x
x=\frac{\sqrt{823241}}{10}-76.4\approx 14.332629191
x=-\frac{\sqrt{823241}}{10}-76.4\approx -167.132629191
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0.0005\left(506.25+45x+x^{2}\right)+0.0539\left(22.5+x\right)-1.5816=1.082
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(22.5+x\right)^{2}.
0.253125+0.0225x+0.0005x^{2}+0.0539\left(22.5+x\right)-1.5816=1.082
Use the distributive property to multiply 0.0005 by 506.25+45x+x^{2}.
0.253125+0.0225x+0.0005x^{2}+1.21275+0.0539x-1.5816=1.082
Use the distributive property to multiply 0.0539 by 22.5+x.
1.465875+0.0225x+0.0005x^{2}+0.0539x-1.5816=1.082
Add 0.253125 and 1.21275 to get 1.465875.
1.465875+0.0764x+0.0005x^{2}-1.5816=1.082
Combine 0.0225x and 0.0539x to get 0.0764x.
-0.115725+0.0764x+0.0005x^{2}=1.082
Subtract 1.5816 from 1.465875 to get -0.115725.
-0.115725+0.0764x+0.0005x^{2}-1.082=0
Subtract 1.082 from both sides.
-1.197725+0.0764x+0.0005x^{2}=0
Subtract 1.082 from -0.115725 to get -1.197725.
0.0005x^{2}+0.0764x-1.197725=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{-0.0764±\sqrt{0.0764^{2}-4\times 0.0005\left(-1.197725\right)}}{2\times 0.0005}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0005 for a, 0.0764 for b, and -1.197725 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-0.0764±\sqrt{0.00583696-4\times 0.0005\left(-1.197725\right)}}{2\times 0.0005}
Square 0.0764 by squaring both the numerator and the denominator of the fraction.
x=\frac{-0.0764±\sqrt{0.00583696-0.002\left(-1.197725\right)}}{2\times 0.0005}
Multiply -4 times 0.0005.
x=\frac{-0.0764±\sqrt{0.00583696+0.00239545}}{2\times 0.0005}
Multiply -0.002 times -1.197725 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{-0.0764±\sqrt{0.00823241}}{2\times 0.0005}
Add 0.00583696 to 0.00239545 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{-0.0764±\frac{\sqrt{823241}}{10000}}{2\times 0.0005}
Take the square root of 0.00823241.
x=\frac{-0.0764±\frac{\sqrt{823241}}{10000}}{0.001}
Multiply 2 times 0.0005.
x=\frac{\frac{\sqrt{823241}}{10000}-\frac{191}{2500}}{0.001}
Now solve the equation x=\frac{-0.0764±\frac{\sqrt{823241}}{10000}}{0.001} when ± is plus. Add -0.0764 to \frac{\sqrt{823241}}{10000}.
x=\frac{\sqrt{823241}}{10}-\frac{382}{5}
Divide -\frac{191}{2500}+\frac{\sqrt{823241}}{10000} by 0.001 by multiplying -\frac{191}{2500}+\frac{\sqrt{823241}}{10000} by the reciprocal of 0.001.
x=\frac{-\frac{\sqrt{823241}}{10000}-\frac{191}{2500}}{0.001}
Now solve the equation x=\frac{-0.0764±\frac{\sqrt{823241}}{10000}}{0.001} when ± is minus. Subtract \frac{\sqrt{823241}}{10000} from -0.0764.
x=-\frac{\sqrt{823241}}{10}-\frac{382}{5}
Divide -\frac{191}{2500}-\frac{\sqrt{823241}}{10000} by 0.001 by multiplying -\frac{191}{2500}-\frac{\sqrt{823241}}{10000} by the reciprocal of 0.001.
x=\frac{\sqrt{823241}}{10}-\frac{382}{5} x=-\frac{\sqrt{823241}}{10}-\frac{382}{5}
The equation is now solved.
0.0005\left(506.25+45x+x^{2}\right)+0.0539\left(22.5+x\right)-1.5816=1.082
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(22.5+x\right)^{2}.
0.253125+0.0225x+0.0005x^{2}+0.0539\left(22.5+x\right)-1.5816=1.082
Use the distributive property to multiply 0.0005 by 506.25+45x+x^{2}.
0.253125+0.0225x+0.0005x^{2}+1.21275+0.0539x-1.5816=1.082
Use the distributive property to multiply 0.0539 by 22.5+x.
1.465875+0.0225x+0.0005x^{2}+0.0539x-1.5816=1.082
Add 0.253125 and 1.21275 to get 1.465875.
1.465875+0.0764x+0.0005x^{2}-1.5816=1.082
Combine 0.0225x and 0.0539x to get 0.0764x.
-0.115725+0.0764x+0.0005x^{2}=1.082
Subtract 1.5816 from 1.465875 to get -0.115725.
0.0764x+0.0005x^{2}=1.082+0.115725
Add 0.115725 to both sides.
0.0764x+0.0005x^{2}=1.197725
Add 1.082 and 0.115725 to get 1.197725.
0.0005x^{2}+0.0764x=1.197725
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{0.0005x^{2}+0.0764x}{0.0005}=\frac{1.197725}{0.0005}
Multiply both sides by 2000.
x^{2}+\frac{0.0764}{0.0005}x=\frac{1.197725}{0.0005}
Dividing by 0.0005 undoes the multiplication by 0.0005.
x^{2}+152.8x=\frac{1.197725}{0.0005}
Divide 0.0764 by 0.0005 by multiplying 0.0764 by the reciprocal of 0.0005.
x^{2}+152.8x=2395.45
Divide 1.197725 by 0.0005 by multiplying 1.197725 by the reciprocal of 0.0005.
x^{2}+152.8x+76.4^{2}=2395.45+76.4^{2}
Divide 152.8, the coefficient of the x term, by 2 to get 76.4. Then add the square of 76.4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+152.8x+5836.96=2395.45+5836.96
Square 76.4 by squaring both the numerator and the denominator of the fraction.
x^{2}+152.8x+5836.96=8232.41
Add 2395.45 to 5836.96 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+76.4\right)^{2}=8232.41
Factor x^{2}+152.8x+5836.96. 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+76.4\right)^{2}}=\sqrt{8232.41}
Take the square root of both sides of the equation.
x+76.4=\frac{\sqrt{823241}}{10} x+76.4=-\frac{\sqrt{823241}}{10}
Simplify.
x=\frac{\sqrt{823241}}{10}-\frac{382}{5} x=-\frac{\sqrt{823241}}{10}-\frac{382}{5}
Subtract 76.4 from both sides of the equation.
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