Solve (60+x)*100=x*(300+x)

$$( 60 + x ) \cdot 100 = x \cdot ( 300 + x )$$

Solve for x

$$x=40\sqrt{10}-100\approx 26.491106407$$

$$x=-40\sqrt{10}-100\approx -226.491106407$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 60+x by 100.

$$6000+100x=x\left(300+x\right)$$

Use the distributive property to multiply x by 300+x.

$$6000+100x=300x+x^{2}$$

Subtract 300x from both sides.

$$6000+100x-300x=x^{2}$$

Combine 100x and -300x to get -200x.

$$6000-200x=x^{2}$$

Subtract x^{2} from both sides.

$$6000-200x-x^{2}=0$$

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.

$$-x^{2}-200x+6000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -200 for b, and 6000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\left(-1\right)\times 6000}}{2\left(-1\right)}$$

Square -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\left(-1\right)\times 6000}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-200\right)±\sqrt{40000+4\times 6000}}{2\left(-1\right)}$$

Multiply 4 times 6000.

$$x=\frac{-\left(-200\right)±\sqrt{40000+24000}}{2\left(-1\right)}$$

Add 40000 to 24000.

$$x=\frac{-\left(-200\right)±\sqrt{64000}}{2\left(-1\right)}$$

Take the square root of 64000.

$$x=\frac{-\left(-200\right)±80\sqrt{10}}{2\left(-1\right)}$$

The opposite of -200 is 200.

$$x=\frac{200±80\sqrt{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{200±80\sqrt{10}}{-2}$$

Now solve the equation x=\frac{200±80\sqrt{10}}{-2} when ± is plus. Add 200 to 80\sqrt{10}.

$$x=\frac{80\sqrt{10}+200}{-2}$$

Divide 200+80\sqrt{10} by -2.

$$x=-40\sqrt{10}-100$$

Now solve the equation x=\frac{200±80\sqrt{10}}{-2} when ± is minus. Subtract 80\sqrt{10} from 200.

$$x=\frac{200-80\sqrt{10}}{-2}$$

Divide 200-80\sqrt{10} by -2.

$$x=40\sqrt{10}-100$$

The equation is now solved.

$$x=-40\sqrt{10}-100 x=40\sqrt{10}-100$$

Steps for Completing the Square

Use the distributive property to multiply 60+x by 100.

$$6000+100x=x\left(300+x\right)$$

Use the distributive property to multiply x by 300+x.

$$6000+100x=300x+x^{2}$$

Subtract 300x from both sides.

$$6000+100x-300x=x^{2}$$

Combine 100x and -300x to get -200x.

$$6000-200x=x^{2}$$

Subtract x^{2} from both sides.

$$6000-200x-x^{2}=0$$

Subtract 6000 from both sides. Anything subtracted from zero gives its negation.

$$-200x-x^{2}=-6000$$

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.

$$-x^{2}-200x=-6000$$

Divide both sides by -1.

$$\frac{-x^{2}-200x}{-1}=-\frac{6000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{200}{-1}\right)x=-\frac{6000}{-1}$$

Divide -200 by -1.

$$x^{2}+200x=-\frac{6000}{-1}$$

Divide -6000 by -1.

$$x^{2}+200x=6000$$

Divide 200, the coefficient of the x term, by 2 to get 100. Then add the square of 100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+200x+100^{2}=6000+100^{2}$$

Square 100.

$$x^{2}+200x+10000=6000+10000$$

Add 6000 to 10000.

$$x^{2}+200x+10000=16000$$

Factor x^{2}+200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+100\right)^{2}=16000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+100\right)^{2}}=\sqrt{16000}$$

Simplify.

$$x+100=40\sqrt{10} x+100=-40\sqrt{10}$$

Subtract 100 from both sides of the equation.

$$x=40\sqrt{10}-100 x=-40\sqrt{10}-100$$