Solve 10x^2-200x=1000

$$10 x ^ { 2 } - 200 x = 1000$$

Solve for x

$$x=10\sqrt{2}+10\approx 24.142135624$$

$$x=10-10\sqrt{2}\approx -4.142135624$$

Steps Using the Quadratic Formula

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.

$$10x^{2}-200x=1000$$

Subtract 1000 from both sides of the equation.

$$10x^{2}-200x-1000=1000-1000$$

Subtracting 1000 from itself leaves 0.

$$10x^{2}-200x-1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -200 for b, and -1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\times 10\left(-1000\right)}}{2\times 10}$$

Square -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\times 10\left(-1000\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-200\right)±\sqrt{40000-40\left(-1000\right)}}{2\times 10}$$

Multiply -40 times -1000.

$$x=\frac{-\left(-200\right)±\sqrt{40000+40000}}{2\times 10}$$

Add 40000 to 40000.

$$x=\frac{-\left(-200\right)±\sqrt{80000}}{2\times 10}$$

Take the square root of 80000.

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

The opposite of -200 is 200.

$$x=\frac{200±200\sqrt{2}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{200±200\sqrt{2}}{20}$$

Now solve the equation x=\frac{200±200\sqrt{2}}{20} when ± is plus. Add 200 to 200\sqrt{2}.

$$x=\frac{200\sqrt{2}+200}{20}$$

Divide 200+200\sqrt{2} by 20.

$$x=10\sqrt{2}+10$$

Now solve the equation x=\frac{200±200\sqrt{2}}{20} when ± is minus. Subtract 200\sqrt{2} from 200.

$$x=\frac{200-200\sqrt{2}}{20}$$

Divide 200-200\sqrt{2} by 20.

$$x=10-10\sqrt{2}$$

The equation is now solved.

$$x=10\sqrt{2}+10 x=10-10\sqrt{2}$$

Steps for Completing the Square

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.

$$10x^{2}-200x=1000$$

Divide both sides by 10.

$$\frac{10x^{2}-200x}{10}=\frac{1000}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{200}{10}\right)x=\frac{1000}{10}$$

Divide -200 by 10.

$$x^{2}-20x=\frac{1000}{10}$$

Divide 1000 by 10.

$$x^{2}-20x=100$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=100+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=100+100$$

Add 100 to 100.

$$x^{2}-20x+100=200$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{200}$$

Simplify.

$$x-10=10\sqrt{2} x-10=-10\sqrt{2}$$

Add 10 to both sides of the equation.

$$x=10\sqrt{2}+10 x=10-10\sqrt{2}$$