Solve (12-x)(20+x)=1750

$$(12-x)(20+x)=1750$$

Solve for x (complex solution)

$$x=-3\sqrt{166}i-4\approx -4-38.65229618i$$

$$x=-4+3\sqrt{166}i\approx -4+38.65229618i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12-x by 20+x and combine like terms.

$$240-8x-x^{2}=1750$$

Subtract 1750 from both sides.

$$240-8x-x^{2}-1750=0$$

Subtract 1750 from 240 to get -1510.

$$-1510-8x-x^{2}=0$$

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.

$$-x^{2}-8x-1510=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and -1510 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\left(-1510\right)}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-1510\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\left(-1510\right)}}{2\left(-1\right)}$$

Multiply 4 times -1510.

$$x=\frac{-\left(-8\right)±\sqrt{64-6040}}{2\left(-1\right)}$$

Add 64 to -6040.

$$x=\frac{-\left(-8\right)±\sqrt{-5976}}{2\left(-1\right)}$$

Take the square root of -5976.

$$x=\frac{-\left(-8\right)±6\sqrt{166}i}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±6\sqrt{166}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±6\sqrt{166}i}{-2}$$

Now solve the equation x=\frac{8±6\sqrt{166}i}{-2} when ± is plus. Add 8 to 6i\sqrt{166}.

$$x=\frac{8+6\sqrt{166}i}{-2}$$

Divide 8+6i\sqrt{166} by -2.

$$x=-3\sqrt{166}i-4$$

Now solve the equation x=\frac{8±6\sqrt{166}i}{-2} when ± is minus. Subtract 6i\sqrt{166} from 8.

$$x=\frac{-6\sqrt{166}i+8}{-2}$$

Divide 8-6i\sqrt{166} by -2.

$$x=-4+3\sqrt{166}i$$

The equation is now solved.

$$x=-3\sqrt{166}i-4 x=-4+3\sqrt{166}i$$

Steps for Completing the Square

Use the distributive property to multiply 12-x by 20+x and combine like terms.

$$240-8x-x^{2}=1750$$

Subtract 240 from both sides.

$$-8x-x^{2}=1750-240$$

Subtract 240 from 1750 to get 1510.

$$-8x-x^{2}=1510$$

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.

$$-x^{2}-8x=1510$$

Divide both sides by -1.

$$\frac{-x^{2}-8x}{-1}=\frac{1510}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=\frac{1510}{-1}$$

Divide -8 by -1.

$$x^{2}+8x=\frac{1510}{-1}$$

Divide 1510 by -1.

$$x^{2}+8x=-1510$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-1510+4^{2}$$

Square 4.

$$x^{2}+8x+16=-1510+16$$

Add -1510 to 16.

$$x^{2}+8x+16=-1494$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=-1494$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{-1494}$$

Simplify.

$$x+4=3\sqrt{166}i x+4=-3\sqrt{166}i$$

Subtract 4 from both sides of the equation.

$$x=-4+3\sqrt{166}i x=-3\sqrt{166}i-4$$