Solve 400x+400x*left(x+1right)/2*12+8/100=16176

$$400x+400 \frac{ x \times \left( x+1 \right) }{ 2 \cdot 12 } + \frac{ 8 }{ 100 } = 16176$$

Solve for x

$$x=\frac{\sqrt{2817013}}{50}-\frac{25}{2}\approx 21.067919209$$

$$x=-\frac{\sqrt{2817013}}{50}-\frac{25}{2}\approx -46.067919209$$

Steps Using the Quadratic Formula

Multiply 2 and 12 to get 24.

$$400x+400\times \frac{x\left(x+1\right)}{24}+\frac{8}{100}=16176$$

Express 400\times \frac{x\left(x+1\right)}{24} as a single fraction.

$$400x+\frac{400x\left(x+1\right)}{24}+\frac{8}{100}=16176$$

Reduce the fraction \frac{8}{100} to lowest terms by extracting and canceling out 4.

$$400x+\frac{400x\left(x+1\right)}{24}+\frac{2}{25}=16176$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 24 and 25 is 600. Multiply \frac{400x\left(x+1\right)}{24} times \frac{25}{25}. Multiply \frac{2}{25} times \frac{24}{24}.

$$400x+\frac{25\times 400x\left(x+1\right)}{600}+\frac{2\times 24}{600}=16176$$

Since \frac{25\times 400x\left(x+1\right)}{600} and \frac{2\times 24}{600} have the same denominator, add them by adding their numerators.

$$400x+\frac{25\times 400x\left(x+1\right)+2\times 24}{600}=16176$$

Do the multiplications in 25\times 400x\left(x+1\right)+2\times 24.

$$400x+\frac{10000x^{2}+10000x+48}{600}=16176$$

Divide each term of 10000x^{2}+10000x+48 by 600 to get \frac{50}{3}x^{2}+\frac{50}{3}x+\frac{2}{25}.

$$400x+\frac{50}{3}x^{2}+\frac{50}{3}x+\frac{2}{25}=16176$$

Combine 400x and \frac{50}{3}x to get \frac{1250}{3}x.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}+\frac{2}{25}=16176$$

Subtract 16176 from both sides.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}+\frac{2}{25}-16176=0$$

Convert 16176 to fraction \frac{404400}{25}.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}+\frac{2}{25}-\frac{404400}{25}=0$$

Since \frac{2}{25} and \frac{404400}{25} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}+\frac{2-404400}{25}=0$$

Subtract 404400 from 2 to get -404398.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}-\frac{404398}{25}=0$$

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.

$$\frac{50}{3}x^{2}+\frac{1250}{3}x-\frac{404398}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{50}{3} for a, \frac{1250}{3} for b, and -\frac{404398}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1250}{3}±\sqrt{\left(\frac{1250}{3}\right)^{2}-4\times \frac{50}{3}\left(-\frac{404398}{25}\right)}}{2\times \frac{50}{3}}$$

Square \frac{1250}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1250}{3}±\sqrt{\frac{1562500}{9}-4\times \frac{50}{3}\left(-\frac{404398}{25}\right)}}{2\times \frac{50}{3}}$$

Multiply -4 times \frac{50}{3}.

$$x=\frac{-\frac{1250}{3}±\sqrt{\frac{1562500}{9}-\frac{200}{3}\left(-\frac{404398}{25}\right)}}{2\times \frac{50}{3}}$$

Multiply -\frac{200}{3} times -\frac{404398}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1250}{3}±\sqrt{\frac{1562500}{9}+\frac{3235184}{3}}}{2\times \frac{50}{3}}$$

Add \frac{1562500}{9} to \frac{3235184}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1250}{3}±\sqrt{\frac{11268052}{9}}}{2\times \frac{50}{3}}$$

Take the square root of \frac{11268052}{9}.

$$x=\frac{-\frac{1250}{3}±\frac{2\sqrt{2817013}}{3}}{2\times \frac{50}{3}}$$

Multiply 2 times \frac{50}{3}.

$$x=\frac{-\frac{1250}{3}±\frac{2\sqrt{2817013}}{3}}{\frac{100}{3}}$$

Now solve the equation x=\frac{-\frac{1250}{3}±\frac{2\sqrt{2817013}}{3}}{\frac{100}{3}} when ± is plus. Add -\frac{1250}{3} to \frac{2\sqrt{2817013}}{3}.

$$x=\frac{2\sqrt{2817013}-1250}{3\times \frac{100}{3}}$$

Divide \frac{-1250+2\sqrt{2817013}}{3} by \frac{100}{3} by multiplying \frac{-1250+2\sqrt{2817013}}{3} by the reciprocal of \frac{100}{3}.

$$x=\frac{\sqrt{2817013}}{50}-\frac{25}{2}$$

Now solve the equation x=\frac{-\frac{1250}{3}±\frac{2\sqrt{2817013}}{3}}{\frac{100}{3}} when ± is minus. Subtract \frac{2\sqrt{2817013}}{3} from -\frac{1250}{3}.

$$x=\frac{-2\sqrt{2817013}-1250}{3\times \frac{100}{3}}$$

Divide \frac{-1250-2\sqrt{2817013}}{3} by \frac{100}{3} by multiplying \frac{-1250-2\sqrt{2817013}}{3} by the reciprocal of \frac{100}{3}.

$$x=-\frac{\sqrt{2817013}}{50}-\frac{25}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2817013}}{50}-\frac{25}{2} x=-\frac{\sqrt{2817013}}{50}-\frac{25}{2}$$

Steps for Completing the Square

Multiply 2 and 12 to get 24.

$$400x+400\times \frac{x\left(x+1\right)}{24}+\frac{8}{100}=16176$$

Express 400\times \frac{x\left(x+1\right)}{24} as a single fraction.

$$400x+\frac{400x\left(x+1\right)}{24}+\frac{8}{100}=16176$$

Reduce the fraction \frac{8}{100} to lowest terms by extracting and canceling out 4.

$$400x+\frac{400x\left(x+1\right)}{24}+\frac{2}{25}=16176$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 24 and 25 is 600. Multiply \frac{400x\left(x+1\right)}{24} times \frac{25}{25}. Multiply \frac{2}{25} times \frac{24}{24}.

$$400x+\frac{25\times 400x\left(x+1\right)}{600}+\frac{2\times 24}{600}=16176$$

Since \frac{25\times 400x\left(x+1\right)}{600} and \frac{2\times 24}{600} have the same denominator, add them by adding their numerators.

$$400x+\frac{25\times 400x\left(x+1\right)+2\times 24}{600}=16176$$

Do the multiplications in 25\times 400x\left(x+1\right)+2\times 24.

$$400x+\frac{10000x^{2}+10000x+48}{600}=16176$$

Divide each term of 10000x^{2}+10000x+48 by 600 to get \frac{50}{3}x^{2}+\frac{50}{3}x+\frac{2}{25}.

$$400x+\frac{50}{3}x^{2}+\frac{50}{3}x+\frac{2}{25}=16176$$

Combine 400x and \frac{50}{3}x to get \frac{1250}{3}x.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}+\frac{2}{25}=16176$$

Subtract \frac{2}{25} from both sides.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}=16176-\frac{2}{25}$$

Convert 16176 to fraction \frac{404400}{25}.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}=\frac{404400}{25}-\frac{2}{25}$$

Since \frac{404400}{25} and \frac{2}{25} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}=\frac{404400-2}{25}$$

Subtract 2 from 404400 to get 404398.

$$\frac{1250}{3}x+\frac{50}{3}x^{2}=\frac{404398}{25}$$

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.

$$\frac{50}{3}x^{2}+\frac{1250}{3}x=\frac{404398}{25}$$

Divide both sides of the equation by \frac{50}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{50}{3}x^{2}+\frac{1250}{3}x}{\frac{50}{3}}=\frac{\frac{404398}{25}}{\frac{50}{3}}$$

Dividing by \frac{50}{3} undoes the multiplication by \frac{50}{3}.

$$x^{2}+\frac{\frac{1250}{3}}{\frac{50}{3}}x=\frac{\frac{404398}{25}}{\frac{50}{3}}$$

Divide \frac{1250}{3} by \frac{50}{3} by multiplying \frac{1250}{3} by the reciprocal of \frac{50}{3}.

$$x^{2}+25x=\frac{\frac{404398}{25}}{\frac{50}{3}}$$

Divide \frac{404398}{25} by \frac{50}{3} by multiplying \frac{404398}{25} by the reciprocal of \frac{50}{3}.

$$x^{2}+25x=\frac{606597}{625}$$

Divide 25, the coefficient of the x term, by 2 to get \frac{25}{2}. Then add the square of \frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+25x+\left(\frac{25}{2}\right)^{2}=\frac{606597}{625}+\left(\frac{25}{2}\right)^{2}$$

Square \frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+25x+\frac{625}{4}=\frac{606597}{625}+\frac{625}{4}$$

Add \frac{606597}{625} to \frac{625}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+25x+\frac{625}{4}=\frac{2817013}{2500}$$

Factor x^{2}+25x+\frac{625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{2}\right)^{2}=\frac{2817013}{2500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{2}\right)^{2}}=\sqrt{\frac{2817013}{2500}}$$

Simplify.

$$x+\frac{25}{2}=\frac{\sqrt{2817013}}{50} x+\frac{25}{2}=-\frac{\sqrt{2817013}}{50}$$

Subtract \frac{25}{2} from both sides of the equation.

$$x=\frac{\sqrt{2817013}}{50}-\frac{25}{2} x=-\frac{\sqrt{2817013}}{50}-\frac{25}{2}$$