Solve 0=60x^2-600x+1000

$$0=60 { x }^{ 2 } -600x+1000$$

Solve for x

$$x=\frac{5\sqrt{3}}{3}+5\approx 7.886751346$$

$$x=-\frac{5\sqrt{3}}{3}+5\approx 2.113248654$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$60x^{2}-600x+1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 60 for a, -600 for b, and 1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-600\right)±\sqrt{\left(-600\right)^{2}-4\times 60\times 1000}}{2\times 60}$$

Square -600.

$$x=\frac{-\left(-600\right)±\sqrt{360000-4\times 60\times 1000}}{2\times 60}$$

Multiply -4 times 60.

$$x=\frac{-\left(-600\right)±\sqrt{360000-240\times 1000}}{2\times 60}$$

Multiply -240 times 1000.

$$x=\frac{-\left(-600\right)±\sqrt{360000-240000}}{2\times 60}$$

Add 360000 to -240000.

$$x=\frac{-\left(-600\right)±\sqrt{120000}}{2\times 60}$$

Take the square root of 120000.

$$x=\frac{-\left(-600\right)±200\sqrt{3}}{2\times 60}$$

The opposite of -600 is 600.

$$x=\frac{600±200\sqrt{3}}{2\times 60}$$

Multiply 2 times 60.

$$x=\frac{600±200\sqrt{3}}{120}$$

Now solve the equation x=\frac{600±200\sqrt{3}}{120} when ± is plus. Add 600 to 200\sqrt{3}.

$$x=\frac{200\sqrt{3}+600}{120}$$

Divide 600+200\sqrt{3} by 120.

$$x=\frac{5\sqrt{3}}{3}+5$$

Now solve the equation x=\frac{600±200\sqrt{3}}{120} when ± is minus. Subtract 200\sqrt{3} from 600.

$$x=\frac{600-200\sqrt{3}}{120}$$

Divide 600-200\sqrt{3} by 120.

$$x=-\frac{5\sqrt{3}}{3}+5$$

The equation is now solved.

$$x=\frac{5\sqrt{3}}{3}+5 x=-\frac{5\sqrt{3}}{3}+5$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$60x^{2}-600x+1000=0$$

Subtract 1000 from both sides. Anything subtracted from zero gives its negation.

$$60x^{2}-600x=-1000$$

Divide both sides by 60.

$$\frac{60x^{2}-600x}{60}=-\frac{1000}{60}$$

Dividing by 60 undoes the multiplication by 60.

$$x^{2}+\left(-\frac{600}{60}\right)x=-\frac{1000}{60}$$

Divide -600 by 60.

$$x^{2}-10x=-\frac{1000}{60}$$

Reduce the fraction \frac{-1000}{60} to lowest terms by extracting and canceling out 20.

$$x^{2}-10x=-\frac{50}{3}$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-\frac{50}{3}+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-\frac{50}{3}+25$$

Add -\frac{50}{3} to 25.

$$x^{2}-10x+25=\frac{25}{3}$$

Factor x^{2}-10x+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}.

$$\left(x-5\right)^{2}=\frac{25}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{\frac{25}{3}}$$

Simplify.

$$x-5=\frac{5\sqrt{3}}{3} x-5=-\frac{5\sqrt{3}}{3}$$

Add 5 to both sides of the equation.

$$x=\frac{5\sqrt{3}}{3}+5 x=-\frac{5\sqrt{3}}{3}+5$$