Solve left(-10xright)^2+4000x-9000=391000

$${ \left(-10x \right) }^{ 2 } +4000x-9000=391000$$

Solve for x

$$x=20\sqrt{11}-20\approx 46.332495807$$

$$x=-20\sqrt{11}-20\approx -86.332495807$$

Steps Using the Quadratic Formula

Expand \left(-10x\right)^{2}.

$$\left(-10\right)^{2}x^{2}+4000x-9000=391000$$

Calculate -10 to the power of 2 and get 100.

$$100x^{2}+4000x-9000=391000$$

Subtract 391000 from both sides.

$$100x^{2}+4000x-9000-391000=0$$

Subtract 391000 from -9000 to get -400000.

$$100x^{2}+4000x-400000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 4000 for b, and -400000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4000±\sqrt{4000^{2}-4\times 100\left(-400000\right)}}{2\times 100}$$

Square 4000.

$$x=\frac{-4000±\sqrt{16000000-4\times 100\left(-400000\right)}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-4000±\sqrt{16000000-400\left(-400000\right)}}{2\times 100}$$

Multiply -400 times -400000.

$$x=\frac{-4000±\sqrt{16000000+160000000}}{2\times 100}$$

Add 16000000 to 160000000.

$$x=\frac{-4000±\sqrt{176000000}}{2\times 100}$$

Take the square root of 176000000.

$$x=\frac{-4000±4000\sqrt{11}}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{-4000±4000\sqrt{11}}{200}$$

Now solve the equation x=\frac{-4000±4000\sqrt{11}}{200} when ± is plus. Add -4000 to 4000\sqrt{11}.

$$x=\frac{4000\sqrt{11}-4000}{200}$$

Divide -4000+4000\sqrt{11} by 200.

$$x=20\sqrt{11}-20$$

Now solve the equation x=\frac{-4000±4000\sqrt{11}}{200} when ± is minus. Subtract 4000\sqrt{11} from -4000.

$$x=\frac{-4000\sqrt{11}-4000}{200}$$

Divide -4000-4000\sqrt{11} by 200.

$$x=-20\sqrt{11}-20$$

The equation is now solved.

$$x=20\sqrt{11}-20 x=-20\sqrt{11}-20$$

Steps for Completing the Square

Expand \left(-10x\right)^{2}.

$$\left(-10\right)^{2}x^{2}+4000x-9000=391000$$

Calculate -10 to the power of 2 and get 100.

$$100x^{2}+4000x-9000=391000$$

Add 9000 to both sides.

$$100x^{2}+4000x=391000+9000$$

Add 391000 and 9000 to get 400000.

$$100x^{2}+4000x=400000$$

Divide both sides by 100.

$$\frac{100x^{2}+4000x}{100}=\frac{400000}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\frac{4000}{100}x=\frac{400000}{100}$$

Divide 4000 by 100.

$$x^{2}+40x=\frac{400000}{100}$$

Divide 400000 by 100.

$$x^{2}+40x=4000$$

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+40x+20^{2}=4000+20^{2}$$

Square 20.

$$x^{2}+40x+400=4000+400$$

Add 4000 to 400.

$$x^{2}+40x+400=4400$$

Factor x^{2}+40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+20\right)^{2}=4400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{4400}$$

Simplify.

$$x+20=20\sqrt{11} x+20=-20\sqrt{11}$$

Subtract 20 from both sides of the equation.

$$x=20\sqrt{11}-20 x=-20\sqrt{11}-20$$