Solve [(1200-x)+50*7+116]x=33000

$$[ ( 1200 - x ) + 50 \times 7 + 116 ] x = 33000$$

Solve for x

$$x=\sqrt{660889}+833\approx 1645.950798019$$

$$x=833-\sqrt{660889}\approx 20.049201981$$

Steps Using the Quadratic Formula

Multiply 50 and 7 to get 350.

$$\left(1200-x+350+116\right)x=33000$$

Add 1200 and 350 to get 1550.

$$\left(1550-x+116\right)x=33000$$

Add 1550 and 116 to get 1666.

$$\left(1666-x\right)x=33000$$

Use the distributive property to multiply 1666-x by x.

$$1666x-x^{2}=33000$$

Subtract 33000 from both sides.

$$1666x-x^{2}-33000=0$$

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.

$$-x^{2}+1666x-33000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1666 for b, and -33000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1666±\sqrt{1666^{2}-4\left(-1\right)\left(-33000\right)}}{2\left(-1\right)}$$

Square 1666.

$$x=\frac{-1666±\sqrt{2775556-4\left(-1\right)\left(-33000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1666±\sqrt{2775556+4\left(-33000\right)}}{2\left(-1\right)}$$

Multiply 4 times -33000.

$$x=\frac{-1666±\sqrt{2775556-132000}}{2\left(-1\right)}$$

Add 2775556 to -132000.

$$x=\frac{-1666±\sqrt{2643556}}{2\left(-1\right)}$$

Take the square root of 2643556.

$$x=\frac{-1666±2\sqrt{660889}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1666±2\sqrt{660889}}{-2}$$

Now solve the equation x=\frac{-1666±2\sqrt{660889}}{-2} when ± is plus. Add -1666 to 2\sqrt{660889}.

$$x=\frac{2\sqrt{660889}-1666}{-2}$$

Divide -1666+2\sqrt{660889} by -2.

$$x=833-\sqrt{660889}$$

Now solve the equation x=\frac{-1666±2\sqrt{660889}}{-2} when ± is minus. Subtract 2\sqrt{660889} from -1666.

$$x=\frac{-2\sqrt{660889}-1666}{-2}$$

Divide -1666-2\sqrt{660889} by -2.

$$x=\sqrt{660889}+833$$

The equation is now solved.

$$x=833-\sqrt{660889} x=\sqrt{660889}+833$$

Steps for Completing the Square

Multiply 50 and 7 to get 350.

$$\left(1200-x+350+116\right)x=33000$$

Add 1200 and 350 to get 1550.

$$\left(1550-x+116\right)x=33000$$

Add 1550 and 116 to get 1666.

$$\left(1666-x\right)x=33000$$

Use the distributive property to multiply 1666-x by x.

$$1666x-x^{2}=33000$$

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.

$$-x^{2}+1666x=33000$$

Divide both sides by -1.

$$\frac{-x^{2}+1666x}{-1}=\frac{33000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1666}{-1}x=\frac{33000}{-1}$$

Divide 1666 by -1.

$$x^{2}-1666x=\frac{33000}{-1}$$

Divide 33000 by -1.

$$x^{2}-1666x=-33000$$

Divide -1666, the coefficient of the x term, by 2 to get -833. Then add the square of -833 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1666x+\left(-833\right)^{2}=-33000+\left(-833\right)^{2}$$

Square -833.

$$x^{2}-1666x+693889=-33000+693889$$

Add -33000 to 693889.

$$x^{2}-1666x+693889=660889$$

Factor x^{2}-1666x+693889. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-833\right)^{2}=660889$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-833\right)^{2}}=\sqrt{660889}$$

Simplify.

$$x-833=\sqrt{660889} x-833=-\sqrt{660889}$$

Add 833 to both sides of the equation.

$$x=\sqrt{660889}+833 x=833-\sqrt{660889}$$