Solve P(x)=300-x/30quad(x)=150,000+3x

$$P ( x ) = 300 - \frac { x } { 30 } \quad ( x ) = 150,000 + 3 x$$

Solve for x (complex solution)

$$x=-45+15\sqrt{19951}i\approx -45+2118.720132533i$$

$$x=-15\sqrt{19951}i-45\approx -45-2118.720132533i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30.

$$9000-xx=4500000+90x$$

Multiply x and x to get x^{2}.

$$9000-x^{2}=4500000+90x$$

Subtract 4500000 from both sides.

$$9000-x^{2}-4500000=90x$$

Subtract 4500000 from 9000 to get -4491000.

$$-4491000-x^{2}=90x$$

Subtract 90x from both sides.

$$-4491000-x^{2}-90x=0$$

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.

$$-x^{2}-90x-4491000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -90 for b, and -4491000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\left(-1\right)\left(-4491000\right)}}{2\left(-1\right)}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\left(-1\right)\left(-4491000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-90\right)±\sqrt{8100+4\left(-4491000\right)}}{2\left(-1\right)}$$

Multiply 4 times -4491000.

$$x=\frac{-\left(-90\right)±\sqrt{8100-17964000}}{2\left(-1\right)}$$

Add 8100 to -17964000.

$$x=\frac{-\left(-90\right)±\sqrt{-17955900}}{2\left(-1\right)}$$

Take the square root of -17955900.

$$x=\frac{-\left(-90\right)±30\sqrt{19951}i}{2\left(-1\right)}$$

The opposite of -90 is 90.

$$x=\frac{90±30\sqrt{19951}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{90±30\sqrt{19951}i}{-2}$$

Now solve the equation x=\frac{90±30\sqrt{19951}i}{-2} when ± is plus. Add 90 to 30i\sqrt{19951}.

$$x=\frac{90+30\sqrt{19951}i}{-2}$$

Divide 90+30i\sqrt{19951} by -2.

$$x=-15\sqrt{19951}i-45$$

Now solve the equation x=\frac{90±30\sqrt{19951}i}{-2} when ± is minus. Subtract 30i\sqrt{19951} from 90.

$$x=\frac{-30\sqrt{19951}i+90}{-2}$$

Divide 90-30i\sqrt{19951} by -2.

$$x=-45+15\sqrt{19951}i$$

The equation is now solved.

$$x=-15\sqrt{19951}i-45 x=-45+15\sqrt{19951}i$$

Steps for Completing the Square

Multiply both sides of the equation by 30.

$$9000-xx=4500000+90x$$

Multiply x and x to get x^{2}.

$$9000-x^{2}=4500000+90x$$

Subtract 90x from both sides.

$$9000-x^{2}-90x=4500000$$

Subtract 9000 from both sides.

$$-x^{2}-90x=4500000-9000$$

Subtract 9000 from 4500000 to get 4491000.

$$-x^{2}-90x=4491000$$

Divide both sides by -1.

$$\frac{-x^{2}-90x}{-1}=\frac{4491000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{90}{-1}\right)x=\frac{4491000}{-1}$$

Divide -90 by -1.

$$x^{2}+90x=\frac{4491000}{-1}$$

Divide 4491000 by -1.

$$x^{2}+90x=-4491000$$

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.

$$x^{2}+90x+45^{2}=-4491000+45^{2}$$

Square 45.

$$x^{2}+90x+2025=-4491000+2025$$

Add -4491000 to 2025.

$$x^{2}+90x+2025=-4488975$$

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}.

$$\left(x+45\right)^{2}=-4488975$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+45\right)^{2}}=\sqrt{-4488975}$$

Simplify.

$$x+45=15\sqrt{19951}i x+45=-15\sqrt{19951}i$$

Subtract 45 from both sides of the equation.

$$x=-45+15\sqrt{19951}i x=-15\sqrt{19951}i-45$$