Solve 2left(65-xright)15-xleft(24-2left(x-5right)right)=550

$$2 \left( 65-x \right) 15-x \left( 24-2 \left( x-5 \right) \right) = 550$$

Solve for x (complex solution)

$$x=16+2\sqrt{111}i\approx 16+21.071307506i$$

$$x=-2\sqrt{111}i+16\approx 16-21.071307506i$$

Steps Using the Quadratic Formula

Multiply 2 and 15 to get 30.

$$30\left(65-x\right)-x\left(24-2\left(x-5\right)\right)=550$$

Use the distributive property to multiply 30 by 65-x.

$$1950-30x-x\left(24-2\left(x-5\right)\right)=550$$

Use the distributive property to multiply -2 by x-5.

$$1950-30x-x\left(24-2x+10\right)=550$$

Add 24 and 10 to get 34.

$$1950-30x-x\left(34-2x\right)=550$$

Use the distributive property to multiply x by 34-2x.

$$1950-30x-\left(34x-2x^{2}\right)=550$$

To find the opposite of 34x-2x^{2}, find the opposite of each term.

$$1950-30x-34x-\left(-2x^{2}\right)=550$$

The opposite of -2x^{2} is 2x^{2}.

$$1950-30x-34x+2x^{2}=550$$

Combine -30x and -34x to get -64x.

$$1950-64x+2x^{2}=550$$

Subtract 550 from both sides.

$$1950-64x+2x^{2}-550=0$$

Subtract 550 from 1950 to get 1400.

$$1400-64x+2x^{2}=0$$

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.

$$2x^{2}-64x+1400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -64 for b, and 1400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 2\times 1400}}{2\times 2}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 2\times 1400}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-64\right)±\sqrt{4096-8\times 1400}}{2\times 2}$$

Multiply -8 times 1400.

$$x=\frac{-\left(-64\right)±\sqrt{4096-11200}}{2\times 2}$$

Add 4096 to -11200.

$$x=\frac{-\left(-64\right)±\sqrt{-7104}}{2\times 2}$$

Take the square root of -7104.

$$x=\frac{-\left(-64\right)±8\sqrt{111}i}{2\times 2}$$

The opposite of -64 is 64.

$$x=\frac{64±8\sqrt{111}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{64±8\sqrt{111}i}{4}$$

Now solve the equation x=\frac{64±8\sqrt{111}i}{4} when ± is plus. Add 64 to 8i\sqrt{111}.

$$x=\frac{64+8\sqrt{111}i}{4}$$

Divide 64+8i\sqrt{111} by 4.

$$x=16+2\sqrt{111}i$$

Now solve the equation x=\frac{64±8\sqrt{111}i}{4} when ± is minus. Subtract 8i\sqrt{111} from 64.

$$x=\frac{-8\sqrt{111}i+64}{4}$$

Divide 64-8i\sqrt{111} by 4.

$$x=-2\sqrt{111}i+16$$

The equation is now solved.

$$x=16+2\sqrt{111}i x=-2\sqrt{111}i+16$$

Steps for Completing the Square

Multiply 2 and 15 to get 30.

$$30\left(65-x\right)-x\left(24-2\left(x-5\right)\right)=550$$

Use the distributive property to multiply 30 by 65-x.

$$1950-30x-x\left(24-2\left(x-5\right)\right)=550$$

Use the distributive property to multiply -2 by x-5.

$$1950-30x-x\left(24-2x+10\right)=550$$

Add 24 and 10 to get 34.

$$1950-30x-x\left(34-2x\right)=550$$

Use the distributive property to multiply x by 34-2x.

$$1950-30x-\left(34x-2x^{2}\right)=550$$

To find the opposite of 34x-2x^{2}, find the opposite of each term.

$$1950-30x-34x-\left(-2x^{2}\right)=550$$

The opposite of -2x^{2} is 2x^{2}.

$$1950-30x-34x+2x^{2}=550$$

Combine -30x and -34x to get -64x.

$$1950-64x+2x^{2}=550$$

Subtract 1950 from both sides.

$$-64x+2x^{2}=550-1950$$

Subtract 1950 from 550 to get -1400.

$$-64x+2x^{2}=-1400$$

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.

$$2x^{2}-64x=-1400$$

Divide both sides by 2.

$$\frac{2x^{2}-64x}{2}=-\frac{1400}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{64}{2}\right)x=-\frac{1400}{2}$$

Divide -64 by 2.

$$x^{2}-32x=-\frac{1400}{2}$$

Divide -1400 by 2.

$$x^{2}-32x=-700$$

Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-32x+\left(-16\right)^{2}=-700+\left(-16\right)^{2}$$

Square -16.

$$x^{2}-32x+256=-700+256$$

Add -700 to 256.

$$x^{2}-32x+256=-444$$

Factor x^{2}-32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-16\right)^{2}=-444$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{-444}$$

Simplify.

$$x-16=2\sqrt{111}i x-16=-2\sqrt{111}i$$

Add 16 to both sides of the equation.

$$x=16+2\sqrt{111}i x=-2\sqrt{111}i+16$$