Solve 1500x-20x^2=27000

$$1500x-20 { x }^{ 2 } =27000$$

Solve for x

$$x=30$$

$$x=45$$

Steps Using the Quadratic Formula

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.

$$-20x^{2}+1500x=27000$$

Subtract 27000 from both sides of the equation.

$$-20x^{2}+1500x-27000=27000-27000$$

Subtracting 27000 from itself leaves 0.

$$-20x^{2}+1500x-27000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 1500 for b, and -27000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1500±\sqrt{1500^{2}-4\left(-20\right)\left(-27000\right)}}{2\left(-20\right)}$$

Square 1500.

$$x=\frac{-1500±\sqrt{2250000-4\left(-20\right)\left(-27000\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-1500±\sqrt{2250000+80\left(-27000\right)}}{2\left(-20\right)}$$

Multiply 80 times -27000.

$$x=\frac{-1500±\sqrt{2250000-2160000}}{2\left(-20\right)}$$

Add 2250000 to -2160000.

$$x=\frac{-1500±\sqrt{90000}}{2\left(-20\right)}$$

Take the square root of 90000.

$$x=\frac{-1500±300}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-1500±300}{-40}$$

Now solve the equation x=\frac{-1500±300}{-40} when ± is plus. Add -1500 to 300.

$$x=-\frac{1200}{-40}$$

Divide -1200 by -40.

$$x=30$$

Now solve the equation x=\frac{-1500±300}{-40} when ± is minus. Subtract 300 from -1500.

$$x=-\frac{1800}{-40}$$

Divide -1800 by -40.

$$x=45$$

The equation is now solved.

$$x=30 x=45$$

Steps for Completing the Square

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.

$$-20x^{2}+1500x=27000$$

Divide both sides by -20.

$$\frac{-20x^{2}+1500x}{-20}=\frac{27000}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{1500}{-20}x=\frac{27000}{-20}$$

Divide 1500 by -20.

$$x^{2}-75x=\frac{27000}{-20}$$

Divide 27000 by -20.

$$x^{2}-75x=-1350$$

Divide -75, the coefficient of the x term, by 2 to get -\frac{75}{2}. Then add the square of -\frac{75}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-75x+\left(-\frac{75}{2}\right)^{2}=-1350+\left(-\frac{75}{2}\right)^{2}$$

Square -\frac{75}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-75x+\frac{5625}{4}=-1350+\frac{5625}{4}$$

Add -1350 to \frac{5625}{4}.

$$x^{2}-75x+\frac{5625}{4}=\frac{225}{4}$$

Factor x^{2}-75x+\frac{5625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{75}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{75}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{75}{2}=\frac{15}{2} x-\frac{75}{2}=-\frac{15}{2}$$

Add \frac{75}{2} to both sides of the equation.

$$x=45 x=30$$