Solve (3000+x)(600-10x)=5500

$$( 3000 + x ) ( 600 - 10 x ) = 5500$$

Solve for x

$$x=5\sqrt{93614}-1470\approx 59.820250879$$

$$x=-5\sqrt{93614}-1470\approx -2999.820250879$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3000+x by 600-10x and combine like terms.

$$1800000-29400x-10x^{2}=5500$$

Subtract 5500 from both sides.

$$1800000-29400x-10x^{2}-5500=0$$

Subtract 5500 from 1800000 to get 1794500.

$$1794500-29400x-10x^{2}=0$$

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.

$$-10x^{2}-29400x+1794500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, -29400 for b, and 1794500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29400\right)±\sqrt{\left(-29400\right)^{2}-4\left(-10\right)\times 1794500}}{2\left(-10\right)}$$

Square -29400.

$$x=\frac{-\left(-29400\right)±\sqrt{864360000-4\left(-10\right)\times 1794500}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-\left(-29400\right)±\sqrt{864360000+40\times 1794500}}{2\left(-10\right)}$$

Multiply 40 times 1794500.

$$x=\frac{-\left(-29400\right)±\sqrt{864360000+71780000}}{2\left(-10\right)}$$

Add 864360000 to 71780000.

$$x=\frac{-\left(-29400\right)±\sqrt{936140000}}{2\left(-10\right)}$$

Take the square root of 936140000.

$$x=\frac{-\left(-29400\right)±100\sqrt{93614}}{2\left(-10\right)}$$

The opposite of -29400 is 29400.

$$x=\frac{29400±100\sqrt{93614}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{29400±100\sqrt{93614}}{-20}$$

Now solve the equation x=\frac{29400±100\sqrt{93614}}{-20} when ± is plus. Add 29400 to 100\sqrt{93614}.

$$x=\frac{100\sqrt{93614}+29400}{-20}$$

Divide 29400+100\sqrt{93614} by -20.

$$x=-5\sqrt{93614}-1470$$

Now solve the equation x=\frac{29400±100\sqrt{93614}}{-20} when ± is minus. Subtract 100\sqrt{93614} from 29400.

$$x=\frac{29400-100\sqrt{93614}}{-20}$$

Divide 29400-100\sqrt{93614} by -20.

$$x=5\sqrt{93614}-1470$$

The equation is now solved.

$$x=-5\sqrt{93614}-1470 x=5\sqrt{93614}-1470$$

Steps for Completing the Square

Use the distributive property to multiply 3000+x by 600-10x and combine like terms.

$$1800000-29400x-10x^{2}=5500$$

Subtract 1800000 from both sides.

$$-29400x-10x^{2}=5500-1800000$$

Subtract 1800000 from 5500 to get -1794500.

$$-29400x-10x^{2}=-1794500$$

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.

$$-10x^{2}-29400x=-1794500$$

Divide both sides by -10.

$$\frac{-10x^{2}-29400x}{-10}=-\frac{1794500}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\left(-\frac{29400}{-10}\right)x=-\frac{1794500}{-10}$$

Divide -29400 by -10.

$$x^{2}+2940x=-\frac{1794500}{-10}$$

Divide -1794500 by -10.

$$x^{2}+2940x=179450$$

Divide 2940, the coefficient of the x term, by 2 to get 1470. Then add the square of 1470 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2940x+1470^{2}=179450+1470^{2}$$

Square 1470.

$$x^{2}+2940x+2160900=179450+2160900$$

Add 179450 to 2160900.

$$x^{2}+2940x+2160900=2340350$$

Factor x^{2}+2940x+2160900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1470\right)^{2}=2340350$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1470\right)^{2}}=\sqrt{2340350}$$

Simplify.

$$x+1470=5\sqrt{93614} x+1470=-5\sqrt{93614}$$

Subtract 1470 from both sides of the equation.

$$x=5\sqrt{93614}-1470 x=-5\sqrt{93614}-1470$$