Solve 300/2*x^2=7.8*4200*(406-x)

$$\frac { 300 } { 2 } \times x ^ { 2 } = 7.8 \times 4200 \times ( 406 - x )$$

Solve for x

$$x = \frac{14 \sqrt{12831} - 546}{5} \approx 207.96721142$$

$$x=\frac{-14\sqrt{12831}-546}{5}\approx -426.36721142$$

Steps Using the Quadratic Formula

Divide 300 by 2 to get 150.

$$150x^{2}=7.8\times 4200\left(406-x\right)$$

Multiply 7.8 and 4200 to get 32760.

$$150x^{2}=32760\left(406-x\right)$$

Use the distributive property to multiply 32760 by 406-x.

$$150x^{2}=13300560-32760x$$

Subtract 13300560 from both sides.

$$150x^{2}-13300560=-32760x$$

Add 32760x to both sides.

$$150x^{2}-13300560+32760x=0$$

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.

$$150x^{2}+32760x-13300560=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 150 for a, 32760 for b, and -13300560 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32760±\sqrt{32760^{2}-4\times 150\left(-13300560\right)}}{2\times 150}$$

Square 32760.

$$x=\frac{-32760±\sqrt{1073217600-4\times 150\left(-13300560\right)}}{2\times 150}$$

Multiply -4 times 150.

$$x=\frac{-32760±\sqrt{1073217600-600\left(-13300560\right)}}{2\times 150}$$

Multiply -600 times -13300560.

$$x=\frac{-32760±\sqrt{1073217600+7980336000}}{2\times 150}$$

Add 1073217600 to 7980336000.

$$x=\frac{-32760±\sqrt{9053553600}}{2\times 150}$$

Take the square root of 9053553600.

$$x=\frac{-32760±840\sqrt{12831}}{2\times 150}$$

Multiply 2 times 150.

$$x=\frac{-32760±840\sqrt{12831}}{300}$$

Now solve the equation x=\frac{-32760±840\sqrt{12831}}{300} when ± is plus. Add -32760 to 840\sqrt{12831}.

$$x=\frac{840\sqrt{12831}-32760}{300}$$

Divide -32760+840\sqrt{12831} by 300.

$$x=\frac{14\sqrt{12831}-546}{5}$$

Now solve the equation x=\frac{-32760±840\sqrt{12831}}{300} when ± is minus. Subtract 840\sqrt{12831} from -32760.

$$x=\frac{-840\sqrt{12831}-32760}{300}$$

Divide -32760-840\sqrt{12831} by 300.

$$x=\frac{-14\sqrt{12831}-546}{5}$$

The equation is now solved.

$$x=\frac{14\sqrt{12831}-546}{5} x=\frac{-14\sqrt{12831}-546}{5}$$

Steps for Completing the Square

Divide 300 by 2 to get 150.

$$150x^{2}=7.8\times 4200\left(406-x\right)$$

Multiply 7.8 and 4200 to get 32760.

$$150x^{2}=32760\left(406-x\right)$$

Use the distributive property to multiply 32760 by 406-x.

$$150x^{2}=13300560-32760x$$

Add 32760x to both sides.

$$150x^{2}+32760x=13300560$$

Divide both sides by 150.

$$\frac{150x^{2}+32760x}{150}=\frac{13300560}{150}$$

Dividing by 150 undoes the multiplication by 150.

$$x^{2}+\frac{32760}{150}x=\frac{13300560}{150}$$

Reduce the fraction \frac{32760}{150} to lowest terms by extracting and canceling out 30.

$$x^{2}+\frac{1092}{5}x=\frac{13300560}{150}$$

Reduce the fraction \frac{13300560}{150} to lowest terms by extracting and canceling out 30.

$$x^{2}+\frac{1092}{5}x=\frac{443352}{5}$$

Divide \frac{1092}{5}, the coefficient of the x term, by 2 to get \frac{546}{5}. Then add the square of \frac{546}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1092}{5}x+\left(\frac{546}{5}\right)^{2}=\frac{443352}{5}+\left(\frac{546}{5}\right)^{2}$$

Square \frac{546}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1092}{5}x+\frac{298116}{25}=\frac{443352}{5}+\frac{298116}{25}$$

Add \frac{443352}{5} to \frac{298116}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1092}{5}x+\frac{298116}{25}=\frac{2514876}{25}$$

Factor x^{2}+\frac{1092}{5}x+\frac{298116}{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}.

$$\left(x+\frac{546}{5}\right)^{2}=\frac{2514876}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{546}{5}\right)^{2}}=\sqrt{\frac{2514876}{25}}$$

Simplify.

$$x+\frac{546}{5}=\frac{14\sqrt{12831}}{5} x+\frac{546}{5}=-\frac{14\sqrt{12831}}{5}$$

Subtract \frac{546}{5} from both sides of the equation.

$$x=\frac{14\sqrt{12831}-546}{5} x=\frac{-14\sqrt{12831}-546}{5}$$