Solve (600+10x)(10+x)=10000

$$(600+10x)(10+x)=10000$$

Solve for x

$$x=5\sqrt{65}-35\approx 5.311288741$$

$$x=-5\sqrt{65}-35\approx -75.311288741$$

Steps Using the Quadratic Formula

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

$$6000+700x+10x^{2}=10000$$

Subtract 10000 from both sides.

$$6000+700x+10x^{2}-10000=0$$

Subtract 10000 from 6000 to get -4000.

$$-4000+700x+10x^{2}=0$$

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.

$$10x^{2}+700x-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 700 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-700±\sqrt{700^{2}-4\times 10\left(-4000\right)}}{2\times 10}$$

Square 700.

$$x=\frac{-700±\sqrt{490000-4\times 10\left(-4000\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-700±\sqrt{490000-40\left(-4000\right)}}{2\times 10}$$

Multiply -40 times -4000.

$$x=\frac{-700±\sqrt{490000+160000}}{2\times 10}$$

Add 490000 to 160000.

$$x=\frac{-700±\sqrt{650000}}{2\times 10}$$

Take the square root of 650000.

$$x=\frac{-700±100\sqrt{65}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-700±100\sqrt{65}}{20}$$

Now solve the equation x=\frac{-700±100\sqrt{65}}{20} when ± is plus. Add -700 to 100\sqrt{65}.

$$x=\frac{100\sqrt{65}-700}{20}$$

Divide -700+100\sqrt{65} by 20.

$$x=5\sqrt{65}-35$$

Now solve the equation x=\frac{-700±100\sqrt{65}}{20} when ± is minus. Subtract 100\sqrt{65} from -700.

$$x=\frac{-100\sqrt{65}-700}{20}$$

Divide -700-100\sqrt{65} by 20.

$$x=-5\sqrt{65}-35$$

The equation is now solved.

$$x=5\sqrt{65}-35 x=-5\sqrt{65}-35$$

Steps for Completing the Square

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

$$6000+700x+10x^{2}=10000$$

Subtract 6000 from both sides.

$$700x+10x^{2}=10000-6000$$

Subtract 6000 from 10000 to get 4000.

$$700x+10x^{2}=4000$$

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.

$$10x^{2}+700x=4000$$

Divide both sides by 10.

$$\frac{10x^{2}+700x}{10}=\frac{4000}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{700}{10}x=\frac{4000}{10}$$

Divide 700 by 10.

$$x^{2}+70x=\frac{4000}{10}$$

Divide 4000 by 10.

$$x^{2}+70x=400$$

Divide 70, the coefficient of the x term, by 2 to get 35. Then add the square of 35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+70x+35^{2}=400+35^{2}$$

Square 35.

$$x^{2}+70x+1225=400+1225$$

Add 400 to 1225.

$$x^{2}+70x+1225=1625$$

Factor x^{2}+70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+35\right)^{2}=1625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+35\right)^{2}}=\sqrt{1625}$$

Simplify.

$$x+35=5\sqrt{65} x+35=-5\sqrt{65}$$

Subtract 35 from both sides of the equation.

$$x=5\sqrt{65}-35 x=-5\sqrt{65}-35$$