Solve 800x-x(-2x+100)=4800

$$800 x - x ( - 2 x + 100 ) = 4800$$

Solve for x

$$x=5\sqrt{1321}-175\approx 6.727818454$$

$$x=-5\sqrt{1321}-175\approx -356.727818454$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by -2x+100.

$$800x-\left(-2x^{2}+100x\right)=4800$$

To find the opposite of -2x^{2}+100x, find the opposite of each term.

$$800x-\left(-2x^{2}\right)-100x=4800$$

The opposite of -2x^{2} is 2x^{2}.

$$800x+2x^{2}-100x=4800$$

Combine 800x and -100x to get 700x.

$$700x+2x^{2}=4800$$

Subtract 4800 from both sides.

$$700x+2x^{2}-4800=0$$

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.

$$2x^{2}+700x-4800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 700 for b, and -4800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-700±\sqrt{700^{2}-4\times 2\left(-4800\right)}}{2\times 2}$$

Square 700.

$$x=\frac{-700±\sqrt{490000-4\times 2\left(-4800\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-700±\sqrt{490000-8\left(-4800\right)}}{2\times 2}$$

Multiply -8 times -4800.

$$x=\frac{-700±\sqrt{490000+38400}}{2\times 2}$$

Add 490000 to 38400.

$$x=\frac{-700±\sqrt{528400}}{2\times 2}$$

Take the square root of 528400.

$$x=\frac{-700±20\sqrt{1321}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-700±20\sqrt{1321}}{4}$$

Now solve the equation x=\frac{-700±20\sqrt{1321}}{4} when ± is plus. Add -700 to 20\sqrt{1321}.

$$x=\frac{20\sqrt{1321}-700}{4}$$

Divide -700+20\sqrt{1321} by 4.

$$x=5\sqrt{1321}-175$$

Now solve the equation x=\frac{-700±20\sqrt{1321}}{4} when ± is minus. Subtract 20\sqrt{1321} from -700.

$$x=\frac{-20\sqrt{1321}-700}{4}$$

Divide -700-20\sqrt{1321} by 4.

$$x=-5\sqrt{1321}-175$$

The equation is now solved.

$$x=5\sqrt{1321}-175 x=-5\sqrt{1321}-175$$

Steps for Completing the Square

Use the distributive property to multiply x by -2x+100.

$$800x-\left(-2x^{2}+100x\right)=4800$$

To find the opposite of -2x^{2}+100x, find the opposite of each term.

$$800x-\left(-2x^{2}\right)-100x=4800$$

The opposite of -2x^{2} is 2x^{2}.

$$800x+2x^{2}-100x=4800$$

Combine 800x and -100x to get 700x.

$$700x+2x^{2}=4800$$

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.

$$2x^{2}+700x=4800$$

Divide both sides by 2.

$$\frac{2x^{2}+700x}{2}=\frac{4800}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{700}{2}x=\frac{4800}{2}$$

Divide 700 by 2.

$$x^{2}+350x=\frac{4800}{2}$$

Divide 4800 by 2.

$$x^{2}+350x=2400$$

Divide 350, the coefficient of the x term, by 2 to get 175. Then add the square of 175 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+350x+175^{2}=2400+175^{2}$$

Square 175.

$$x^{2}+350x+30625=2400+30625$$

Add 2400 to 30625.

$$x^{2}+350x+30625=33025$$

Factor x^{2}+350x+30625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+175\right)^{2}=33025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+175\right)^{2}}=\sqrt{33025}$$

Simplify.

$$x+175=5\sqrt{1321} x+175=-5\sqrt{1321}$$

Subtract 175 from both sides of the equation.

$$x=5\sqrt{1321}-175 x=-5\sqrt{1321}-175$$