Solve [100-(x-40)*10]*(x-40)=8000

$$[ 100 - ( x - 40 ) \times 10 ] \times ( x - 40 ) = 8000$$

Solve for x (complex solution)

$$x=45+5\sqrt{31}i\approx 45+27.838821814i$$

$$x=-5\sqrt{31}i+45\approx 45-27.838821814i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-40 by 10.

$$\left(100-\left(10x-400\right)\right)\left(x-40\right)=8000$$

To find the opposite of 10x-400, find the opposite of each term.

$$\left(100-10x-\left(-400\right)\right)\left(x-40\right)=8000$$

The opposite of -400 is 400.

$$\left(100-10x+400\right)\left(x-40\right)=8000$$

Add 100 and 400 to get 500.

$$\left(500-10x\right)\left(x-40\right)=8000$$

Apply the distributive property by multiplying each term of 500-10x by each term of x-40.

$$500x-20000-10x^{2}+400x=8000$$

Combine 500x and 400x to get 900x.

$$900x-20000-10x^{2}=8000$$

Subtract 8000 from both sides.

$$900x-20000-10x^{2}-8000=0$$

Subtract 8000 from -20000 to get -28000.

$$900x-28000-10x^{2}=0$$

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.

$$-10x^{2}+900x-28000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 900 for b, and -28000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-900±\sqrt{900^{2}-4\left(-10\right)\left(-28000\right)}}{2\left(-10\right)}$$

Square 900.

$$x=\frac{-900±\sqrt{810000-4\left(-10\right)\left(-28000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-900±\sqrt{810000+40\left(-28000\right)}}{2\left(-10\right)}$$

Multiply 40 times -28000.

$$x=\frac{-900±\sqrt{810000-1120000}}{2\left(-10\right)}$$

Add 810000 to -1120000.

$$x=\frac{-900±\sqrt{-310000}}{2\left(-10\right)}$$

Take the square root of -310000.

$$x=\frac{-900±100\sqrt{31}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-900±100\sqrt{31}i}{-20}$$

Now solve the equation x=\frac{-900±100\sqrt{31}i}{-20} when ± is plus. Add -900 to 100i\sqrt{31}.

$$x=\frac{-900+100\sqrt{31}i}{-20}$$

Divide -900+100i\sqrt{31} by -20.

$$x=-5\sqrt{31}i+45$$

Now solve the equation x=\frac{-900±100\sqrt{31}i}{-20} when ± is minus. Subtract 100i\sqrt{31} from -900.

$$x=\frac{-100\sqrt{31}i-900}{-20}$$

Divide -900-100i\sqrt{31} by -20.

$$x=45+5\sqrt{31}i$$

The equation is now solved.

$$x=-5\sqrt{31}i+45 x=45+5\sqrt{31}i$$

Steps for Completing the Square

Use the distributive property to multiply x-40 by 10.

$$\left(100-\left(10x-400\right)\right)\left(x-40\right)=8000$$

To find the opposite of 10x-400, find the opposite of each term.

$$\left(100-10x-\left(-400\right)\right)\left(x-40\right)=8000$$

The opposite of -400 is 400.

$$\left(100-10x+400\right)\left(x-40\right)=8000$$

Add 100 and 400 to get 500.

$$\left(500-10x\right)\left(x-40\right)=8000$$

Apply the distributive property by multiplying each term of 500-10x by each term of x-40.

$$500x-20000-10x^{2}+400x=8000$$

Combine 500x and 400x to get 900x.

$$900x-20000-10x^{2}=8000$$

Add 20000 to both sides.

$$900x-10x^{2}=8000+20000$$

Add 8000 and 20000 to get 28000.

$$900x-10x^{2}=28000$$

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.

$$-10x^{2}+900x=28000$$

Divide both sides by -10.

$$\frac{-10x^{2}+900x}{-10}=\frac{28000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{900}{-10}x=\frac{28000}{-10}$$

Divide 900 by -10.

$$x^{2}-90x=\frac{28000}{-10}$$

Divide 28000 by -10.

$$x^{2}-90x=-2800$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-2800+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-2800+2025$$

Add -2800 to 2025.

$$x^{2}-90x+2025=-775$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=-775$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{-775}$$

Simplify.

$$x-45=5\sqrt{31}i x-45=-5\sqrt{31}i$$

Add 45 to both sides of the equation.

$$x=45+5\sqrt{31}i x=-5\sqrt{31}i+45$$