Solve for x
x = -\frac{17}{5} = -3\frac{2}{5} = -3.4
x=22
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x\times 462-\left(x-1\right)\times 374=5x\left(x-1\right)
Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x-1,x.
x\times 462-\left(374x-374\right)=5x\left(x-1\right)
Use the distributive property to multiply x-1 by 374.
x\times 462-374x+374=5x\left(x-1\right)
To find the opposite of 374x-374, find the opposite of each term.
88x+374=5x\left(x-1\right)
Combine x\times 462 and -374x to get 88x.
88x+374=5x^{2}-5x
Use the distributive property to multiply 5x by x-1.
88x+374-5x^{2}=-5x
Subtract 5x^{2} from both sides.
88x+374-5x^{2}+5x=0
Add 5x to both sides.
93x+374-5x^{2}=0
Combine 88x and 5x to get 93x.
-5x^{2}+93x+374=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=93 ab=-5\times 374=-1870
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx+374. To find a and b, set up a system to be solved.
-1,1870 -2,935 -5,374 -10,187 -11,170 -17,110 -22,85 -34,55
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1870.
-1+1870=1869 -2+935=933 -5+374=369 -10+187=177 -11+170=159 -17+110=93 -22+85=63 -34+55=21
Calculate the sum for each pair.
a=110 b=-17
The solution is the pair that gives sum 93.
\left(-5x^{2}+110x\right)+\left(-17x+374\right)
Rewrite -5x^{2}+93x+374 as \left(-5x^{2}+110x\right)+\left(-17x+374\right).
5x\left(-x+22\right)+17\left(-x+22\right)
Factor out 5x in the first and 17 in the second group.
\left(-x+22\right)\left(5x+17\right)
Factor out common term -x+22 by using distributive property.
x=22 x=-\frac{17}{5}
To find equation solutions, solve -x+22=0 and 5x+17=0.
x\times 462-\left(x-1\right)\times 374=5x\left(x-1\right)
Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x-1,x.
x\times 462-\left(374x-374\right)=5x\left(x-1\right)
Use the distributive property to multiply x-1 by 374.
x\times 462-374x+374=5x\left(x-1\right)
To find the opposite of 374x-374, find the opposite of each term.
88x+374=5x\left(x-1\right)
Combine x\times 462 and -374x to get 88x.
88x+374=5x^{2}-5x
Use the distributive property to multiply 5x by x-1.
88x+374-5x^{2}=-5x
Subtract 5x^{2} from both sides.
88x+374-5x^{2}+5x=0
Add 5x to both sides.
93x+374-5x^{2}=0
Combine 88x and 5x to get 93x.
-5x^{2}+93x+374=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{-93±\sqrt{93^{2}-4\left(-5\right)\times 374}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 93 for b, and 374 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-93±\sqrt{8649-4\left(-5\right)\times 374}}{2\left(-5\right)}
Square 93.
x=\frac{-93±\sqrt{8649+20\times 374}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-93±\sqrt{8649+7480}}{2\left(-5\right)}
Multiply 20 times 374.
x=\frac{-93±\sqrt{16129}}{2\left(-5\right)}
Add 8649 to 7480.
x=\frac{-93±127}{2\left(-5\right)}
Take the square root of 16129.
x=\frac{-93±127}{-10}
Multiply 2 times -5.
x=\frac{34}{-10}
Now solve the equation x=\frac{-93±127}{-10} when ± is plus. Add -93 to 127.
x=-\frac{17}{5}
Reduce the fraction \frac{34}{-10} to lowest terms by extracting and canceling out 2.
x=-\frac{220}{-10}
Now solve the equation x=\frac{-93±127}{-10} when ± is minus. Subtract 127 from -93.
x=22
Divide -220 by -10.
x=-\frac{17}{5} x=22
The equation is now solved.
x\times 462-\left(x-1\right)\times 374=5x\left(x-1\right)
Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x-1,x.
x\times 462-\left(374x-374\right)=5x\left(x-1\right)
Use the distributive property to multiply x-1 by 374.
x\times 462-374x+374=5x\left(x-1\right)
To find the opposite of 374x-374, find the opposite of each term.
88x+374=5x\left(x-1\right)
Combine x\times 462 and -374x to get 88x.
88x+374=5x^{2}-5x
Use the distributive property to multiply 5x by x-1.
88x+374-5x^{2}=-5x
Subtract 5x^{2} from both sides.
88x+374-5x^{2}+5x=0
Add 5x to both sides.
93x+374-5x^{2}=0
Combine 88x and 5x to get 93x.
93x-5x^{2}=-374
Subtract 374 from both sides. Anything subtracted from zero gives its negation.
-5x^{2}+93x=-374
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{-5x^{2}+93x}{-5}=-\frac{374}{-5}
Divide both sides by -5.
x^{2}+\frac{93}{-5}x=-\frac{374}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}-\frac{93}{5}x=-\frac{374}{-5}
Divide 93 by -5.
x^{2}-\frac{93}{5}x=\frac{374}{5}
Divide -374 by -5.
x^{2}-\frac{93}{5}x+\left(-\frac{93}{10}\right)^{2}=\frac{374}{5}+\left(-\frac{93}{10}\right)^{2}
Divide -\frac{93}{5}, the coefficient of the x term, by 2 to get -\frac{93}{10}. Then add the square of -\frac{93}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{93}{5}x+\frac{8649}{100}=\frac{374}{5}+\frac{8649}{100}
Square -\frac{93}{10} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{93}{5}x+\frac{8649}{100}=\frac{16129}{100}
Add \frac{374}{5} to \frac{8649}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{93}{10}\right)^{2}=\frac{16129}{100}
Factor x^{2}-\frac{93}{5}x+\frac{8649}{100}. 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{93}{10}\right)^{2}}=\sqrt{\frac{16129}{100}}
Take the square root of both sides of the equation.
x-\frac{93}{10}=\frac{127}{10} x-\frac{93}{10}=-\frac{127}{10}
Simplify.
x=22 x=-\frac{17}{5}
Add \frac{93}{10} to both sides of the equation.
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