Solve x=8500+10x/1581-31x

$$x = \frac { 8500 + 10 x } { 1581 - 31 x }$$

Solve for x

$$x = \frac{\sqrt{1414041} + 1571}{62} \approx 44.518299343$$

$$x = \frac{1571 - \sqrt{1414041}}{62} \approx 6.159120012$$

Steps Using the Quadratic Formula

Subtract \frac{8500+10x}{1581-31x} from both sides.

$$x-\frac{8500+10x}{1581-31x}=0$$

Factor 1581-31x.

$$x-\frac{8500+10x}{31\left(-x+51\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{31\left(-x+51\right)}{31\left(-x+51\right)}.

$$\frac{x\times 31\left(-x+51\right)}{31\left(-x+51\right)}-\frac{8500+10x}{31\left(-x+51\right)}=0$$

Since \frac{x\times 31\left(-x+51\right)}{31\left(-x+51\right)} and \frac{8500+10x}{31\left(-x+51\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 31\left(-x+51\right)-\left(8500+10x\right)}{31\left(-x+51\right)}=0$$

Do the multiplications in x\times 31\left(-x+51\right)-\left(8500+10x\right).

$$\frac{-31x^{2}+1581x-8500-10x}{31\left(-x+51\right)}=0$$

Combine like terms in -31x^{2}+1581x-8500-10x.

$$\frac{-31x^{2}+1571x-8500}{31\left(-x+51\right)}=0$$

Factor the expressions that are not already factored in \frac{-31x^{2}+1571x-8500}{31\left(-x+51\right)}.

$$\frac{-31\left(x-\left(-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)}{31\left(-x+51\right)}=0$$

Cancel out 31 in both numerator and denominator.

$$\frac{-\left(x-\left(-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)}{-x+51}=0$$

Variable x cannot be equal to 51 since division by zero is not defined. Multiply both sides of the equation by -x+51.

$$-\left(x-\left(-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)=0$$

To find the opposite of -\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}, find the opposite of each term.

$$-\left(x+\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)=0$$

To find the opposite of \frac{1}{62}\sqrt{1414041}+\frac{1571}{62}, find the opposite of each term.

$$-\left(x+\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)\left(x-\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)=0$$

Use the distributive property to multiply -1 by x+\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}.

$$\left(-x-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\left(x-\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)=0$$

Use the distributive property to multiply -x-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62} by x-\frac{1}{62}\sqrt{1414041}-\frac{1571}{62} and combine like terms.

$$-x^{2}+\frac{1571}{31}x+\frac{1}{3844}\left(\sqrt{1414041}\right)^{2}-\frac{2468041}{3844}=0$$

The square of \sqrt{1414041} is 1414041.

$$-x^{2}+\frac{1571}{31}x+\frac{1}{3844}\times 1414041-\frac{2468041}{3844}=0$$

Multiply \frac{1}{3844} and 1414041 to get \frac{1414041}{3844}.

$$-x^{2}+\frac{1571}{31}x+\frac{1414041}{3844}-\frac{2468041}{3844}=0$$

Subtract \frac{2468041}{3844} from \frac{1414041}{3844} to get -\frac{8500}{31}.

$$-x^{2}+\frac{1571}{31}x-\frac{8500}{31}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{1571}{31} for b, and -\frac{8500}{31} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1571}{31}±\sqrt{\left(\frac{1571}{31}\right)^{2}-4\left(-1\right)\left(-\frac{8500}{31}\right)}}{2\left(-1\right)}$$

Square \frac{1571}{31} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1571}{31}±\sqrt{\frac{2468041}{961}-4\left(-1\right)\left(-\frac{8500}{31}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{1571}{31}±\sqrt{\frac{2468041}{961}+4\left(-\frac{8500}{31}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{8500}{31}.

$$x=\frac{-\frac{1571}{31}±\sqrt{\frac{2468041}{961}-\frac{34000}{31}}}{2\left(-1\right)}$$

Add \frac{2468041}{961} to -\frac{34000}{31} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1571}{31}±\sqrt{\frac{1414041}{961}}}{2\left(-1\right)}$$

Take the square root of \frac{1414041}{961}.

$$x=\frac{-\frac{1571}{31}±\frac{\sqrt{1414041}}{31}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{1571}{31}±\frac{\sqrt{1414041}}{31}}{-2}$$

Now solve the equation x=\frac{-\frac{1571}{31}±\frac{\sqrt{1414041}}{31}}{-2} when ± is plus. Add -\frac{1571}{31} to \frac{\sqrt{1414041}}{31}.

$$x=\frac{\sqrt{1414041}-1571}{-2\times 31}$$

Divide \frac{-1571+\sqrt{1414041}}{31} by -2.

$$x=\frac{1571-\sqrt{1414041}}{62}$$

Now solve the equation x=\frac{-\frac{1571}{31}±\frac{\sqrt{1414041}}{31}}{-2} when ± is minus. Subtract \frac{\sqrt{1414041}}{31} from -\frac{1571}{31}.

$$x=\frac{-\sqrt{1414041}-1571}{-2\times 31}$$

Divide \frac{-1571-\sqrt{1414041}}{31} by -2.

$$x=\frac{\sqrt{1414041}+1571}{62}$$

The equation is now solved.

$$x=\frac{1571-\sqrt{1414041}}{62} x=\frac{\sqrt{1414041}+1571}{62}$$

Steps for Completing the Square

Subtract \frac{8500+10x}{1581-31x} from both sides.

$$x-\frac{8500+10x}{1581-31x}=0$$

Factor 1581-31x.

$$x-\frac{8500+10x}{31\left(-x+51\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{31\left(-x+51\right)}{31\left(-x+51\right)}.

$$\frac{x\times 31\left(-x+51\right)}{31\left(-x+51\right)}-\frac{8500+10x}{31\left(-x+51\right)}=0$$

Since \frac{x\times 31\left(-x+51\right)}{31\left(-x+51\right)} and \frac{8500+10x}{31\left(-x+51\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 31\left(-x+51\right)-\left(8500+10x\right)}{31\left(-x+51\right)}=0$$

Do the multiplications in x\times 31\left(-x+51\right)-\left(8500+10x\right).

$$\frac{-31x^{2}+1581x-8500-10x}{31\left(-x+51\right)}=0$$

Combine like terms in -31x^{2}+1581x-8500-10x.

$$\frac{-31x^{2}+1571x-8500}{31\left(-x+51\right)}=0$$

Factor the expressions that are not already factored in \frac{-31x^{2}+1571x-8500}{31\left(-x+51\right)}.

$$\frac{-31\left(x-\left(-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)}{31\left(-x+51\right)}=0$$

Cancel out 31 in both numerator and denominator.

$$\frac{-\left(x-\left(-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)}{-x+51}=0$$

Variable x cannot be equal to 51 since division by zero is not defined. Multiply both sides of the equation by -x+51.

$$-\left(x-\left(-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)=0$$

To find the opposite of -\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}, find the opposite of each term.

$$-\left(x+\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)\left(x-\left(\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\right)=0$$

To find the opposite of \frac{1}{62}\sqrt{1414041}+\frac{1571}{62}, find the opposite of each term.

$$-\left(x+\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)\left(x-\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)=0$$

Use the distributive property to multiply -1 by x+\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}.

$$\left(-x-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62}\right)\left(x-\frac{1}{62}\sqrt{1414041}-\frac{1571}{62}\right)=0$$

Use the distributive property to multiply -x-\frac{1}{62}\sqrt{1414041}+\frac{1571}{62} by x-\frac{1}{62}\sqrt{1414041}-\frac{1571}{62} and combine like terms.

$$-x^{2}+\frac{1571}{31}x+\frac{1}{3844}\left(\sqrt{1414041}\right)^{2}-\frac{2468041}{3844}=0$$

The square of \sqrt{1414041} is 1414041.

$$-x^{2}+\frac{1571}{31}x+\frac{1}{3844}\times 1414041-\frac{2468041}{3844}=0$$

Multiply \frac{1}{3844} and 1414041 to get \frac{1414041}{3844}.

$$-x^{2}+\frac{1571}{31}x+\frac{1414041}{3844}-\frac{2468041}{3844}=0$$

Subtract \frac{2468041}{3844} from \frac{1414041}{3844} to get -\frac{8500}{31}.

$$-x^{2}+\frac{1571}{31}x-\frac{8500}{31}=0$$

Add \frac{8500}{31} to both sides. Anything plus zero gives itself.

$$-x^{2}+\frac{1571}{31}x=\frac{8500}{31}$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{1571}{31}x}{-1}=\frac{\frac{8500}{31}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{1571}{31}}{-1}x=\frac{\frac{8500}{31}}{-1}$$

Divide \frac{1571}{31} by -1.

$$x^{2}-\frac{1571}{31}x=\frac{\frac{8500}{31}}{-1}$$

Divide \frac{8500}{31} by -1.

$$x^{2}-\frac{1571}{31}x=-\frac{8500}{31}$$

Divide -\frac{1571}{31}, the coefficient of the x term, by 2 to get -\frac{1571}{62}. Then add the square of -\frac{1571}{62} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1571}{31}x+\left(-\frac{1571}{62}\right)^{2}=-\frac{8500}{31}+\left(-\frac{1571}{62}\right)^{2}$$

Square -\frac{1571}{62} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1571}{31}x+\frac{2468041}{3844}=-\frac{8500}{31}+\frac{2468041}{3844}$$

Add -\frac{8500}{31} to \frac{2468041}{3844} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1571}{31}x+\frac{2468041}{3844}=\frac{1414041}{3844}$$

Factor x^{2}-\frac{1571}{31}x+\frac{2468041}{3844}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1571}{62}\right)^{2}=\frac{1414041}{3844}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1571}{62}\right)^{2}}=\sqrt{\frac{1414041}{3844}}$$

Simplify.

$$x-\frac{1571}{62}=\frac{\sqrt{1414041}}{62} x-\frac{1571}{62}=-\frac{\sqrt{1414041}}{62}$$

Add \frac{1571}{62} to both sides of the equation.

$$x=\frac{\sqrt{1414041}+1571}{62} x=\frac{1571-\sqrt{1414041}}{62}$$