Solve x/2(2*6950+(x-1)*69500)=165600

$$\frac{ x }{ 2 } (2 \times 6950+(x-1) \times 69500)=165600$$

Solve for x

$$x=\frac{2\sqrt{594781}}{695}+\frac{2}{5}\approx 2.619339457$$

$$x=-\frac{2\sqrt{594781}}{695}+\frac{2}{5}\approx -1.819339457$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(2\times 6950+\left(x-1\right)\times 69500\right)=331200$$

Multiply 2 and 6950 to get 13900.

$$x\left(13900+\left(x-1\right)\times 69500\right)=331200$$

Use the distributive property to multiply x-1 by 69500.

$$x\left(13900+69500x-69500\right)=331200$$

Subtract 69500 from 13900 to get -55600.

$$x\left(-55600+69500x\right)=331200$$

Use the distributive property to multiply x by -55600+69500x.

$$-55600x+69500x^{2}=331200$$

Subtract 331200 from both sides.

$$-55600x+69500x^{2}-331200=0$$

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.

$$69500x^{2}-55600x-331200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 69500 for a, -55600 for b, and -331200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55600\right)±\sqrt{\left(-55600\right)^{2}-4\times 69500\left(-331200\right)}}{2\times 69500}$$

Square -55600.

$$x=\frac{-\left(-55600\right)±\sqrt{3091360000-4\times 69500\left(-331200\right)}}{2\times 69500}$$

Multiply -4 times 69500.

$$x=\frac{-\left(-55600\right)±\sqrt{3091360000-278000\left(-331200\right)}}{2\times 69500}$$

Multiply -278000 times -331200.

$$x=\frac{-\left(-55600\right)±\sqrt{3091360000+92073600000}}{2\times 69500}$$

Add 3091360000 to 92073600000.

$$x=\frac{-\left(-55600\right)±\sqrt{95164960000}}{2\times 69500}$$

Take the square root of 95164960000.

$$x=\frac{-\left(-55600\right)±400\sqrt{594781}}{2\times 69500}$$

The opposite of -55600 is 55600.

$$x=\frac{55600±400\sqrt{594781}}{2\times 69500}$$

Multiply 2 times 69500.

$$x=\frac{55600±400\sqrt{594781}}{139000}$$

Now solve the equation x=\frac{55600±400\sqrt{594781}}{139000} when ± is plus. Add 55600 to 400\sqrt{594781}.

$$x=\frac{400\sqrt{594781}+55600}{139000}$$

Divide 55600+400\sqrt{594781} by 139000.

$$x=\frac{2\sqrt{594781}}{695}+\frac{2}{5}$$

Now solve the equation x=\frac{55600±400\sqrt{594781}}{139000} when ± is minus. Subtract 400\sqrt{594781} from 55600.

$$x=\frac{55600-400\sqrt{594781}}{139000}$$

Divide 55600-400\sqrt{594781} by 139000.

$$x=-\frac{2\sqrt{594781}}{695}+\frac{2}{5}$$

The equation is now solved.

$$x=\frac{2\sqrt{594781}}{695}+\frac{2}{5} x=-\frac{2\sqrt{594781}}{695}+\frac{2}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(2\times 6950+\left(x-1\right)\times 69500\right)=331200$$

Multiply 2 and 6950 to get 13900.

$$x\left(13900+\left(x-1\right)\times 69500\right)=331200$$

Use the distributive property to multiply x-1 by 69500.

$$x\left(13900+69500x-69500\right)=331200$$

Subtract 69500 from 13900 to get -55600.

$$x\left(-55600+69500x\right)=331200$$

Use the distributive property to multiply x by -55600+69500x.

$$-55600x+69500x^{2}=331200$$

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.

$$69500x^{2}-55600x=331200$$

Divide both sides by 69500.

$$\frac{69500x^{2}-55600x}{69500}=\frac{331200}{69500}$$

Dividing by 69500 undoes the multiplication by 69500.

$$x^{2}+\left(-\frac{55600}{69500}\right)x=\frac{331200}{69500}$$

Reduce the fraction \frac{-55600}{69500} to lowest terms by extracting and canceling out 13900.

$$x^{2}-\frac{4}{5}x=\frac{331200}{69500}$$

Reduce the fraction \frac{331200}{69500} to lowest terms by extracting and canceling out 100.

$$x^{2}-\frac{4}{5}x=\frac{3312}{695}$$

Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{5}x+\left(-\frac{2}{5}\right)^{2}=\frac{3312}{695}+\left(-\frac{2}{5}\right)^{2}$$

Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=\frac{3312}{695}+\frac{4}{25}$$

Add \frac{3312}{695} to \frac{4}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=\frac{17116}{3475}$$

Factor x^{2}-\frac{4}{5}x+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{5}\right)^{2}=\frac{17116}{3475}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{5}\right)^{2}}=\sqrt{\frac{17116}{3475}}$$

Simplify.

$$x-\frac{2}{5}=\frac{2\sqrt{594781}}{695} x-\frac{2}{5}=-\frac{2\sqrt{594781}}{695}$$

Add \frac{2}{5} to both sides of the equation.

$$x=\frac{2\sqrt{594781}}{695}+\frac{2}{5} x=-\frac{2\sqrt{594781}}{695}+\frac{2}{5}$$