Solve [1000-20(x-25)]*x=27000

$$[ 1000 - 20 ( x - 25 ) ] \cdot x = 27000$$

Solve for x

$$x=30$$

$$x=45$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -20 by x-25.

$$\left(1000-20x+500\right)x=27000$$

Add 1000 and 500 to get 1500.

$$\left(1500-20x\right)x=27000$$

Use the distributive property to multiply 1500-20x by x.

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

Subtract 27000 from both sides.

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

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.

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

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

Use the distributive property to multiply -20 by x-25.

$$\left(1000-20x+500\right)x=27000$$

Add 1000 and 500 to get 1500.

$$\left(1500-20x\right)x=27000$$

Use the distributive property to multiply 1500-20x by x.

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

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.

$$x=45 x=30$$