Solve 1000=left(10+xright)left(85+2xright)

$$1000 = \left( 10+x \right) \left( 85+2x \right)$$

Solve for x

$$x = \frac{5 \sqrt{489} - 105}{4} \approx 1.391680484$$

$$x=\frac{-5\sqrt{489}-105}{4}\approx -53.891680484$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+x by 85+2x and combine like terms.

$$1000=850+105x+2x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$850+105x+2x^{2}=1000$$

Subtract 1000 from both sides.

$$850+105x+2x^{2}-1000=0$$

Subtract 1000 from 850 to get -150.

$$-150+105x+2x^{2}=0$$

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.

$$2x^{2}+105x-150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 105 for b, and -150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-105±\sqrt{105^{2}-4\times 2\left(-150\right)}}{2\times 2}$$

Square 105.

$$x=\frac{-105±\sqrt{11025-4\times 2\left(-150\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-105±\sqrt{11025-8\left(-150\right)}}{2\times 2}$$

Multiply -8 times -150.

$$x=\frac{-105±\sqrt{11025+1200}}{2\times 2}$$

Add 11025 to 1200.

$$x=\frac{-105±\sqrt{12225}}{2\times 2}$$

Take the square root of 12225.

$$x=\frac{-105±5\sqrt{489}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-105±5\sqrt{489}}{4}$$

Now solve the equation x=\frac{-105±5\sqrt{489}}{4} when ± is plus. Add -105 to 5\sqrt{489}.

$$x=\frac{5\sqrt{489}-105}{4}$$

Now solve the equation x=\frac{-105±5\sqrt{489}}{4} when ± is minus. Subtract 5\sqrt{489} from -105.

$$x=\frac{-5\sqrt{489}-105}{4}$$

The equation is now solved.

$$x=\frac{5\sqrt{489}-105}{4} x=\frac{-5\sqrt{489}-105}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 10+x by 85+2x and combine like terms.

$$1000=850+105x+2x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$850+105x+2x^{2}=1000$$

Subtract 850 from both sides.

$$105x+2x^{2}=1000-850$$

Subtract 850 from 1000 to get 150.

$$105x+2x^{2}=150$$

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.

$$2x^{2}+105x=150$$

Divide both sides by 2.

$$\frac{2x^{2}+105x}{2}=\frac{150}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{105}{2}x=\frac{150}{2}$$

Divide 150 by 2.

$$x^{2}+\frac{105}{2}x=75$$

Divide \frac{105}{2}, the coefficient of the x term, by 2 to get \frac{105}{4}. Then add the square of \frac{105}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{105}{2}x+\left(\frac{105}{4}\right)^{2}=75+\left(\frac{105}{4}\right)^{2}$$

Square \frac{105}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{105}{2}x+\frac{11025}{16}=75+\frac{11025}{16}$$

Add 75 to \frac{11025}{16}.

$$x^{2}+\frac{105}{2}x+\frac{11025}{16}=\frac{12225}{16}$$

Factor x^{2}+\frac{105}{2}x+\frac{11025}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{105}{4}\right)^{2}=\frac{12225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{105}{4}\right)^{2}}=\sqrt{\frac{12225}{16}}$$

Simplify.

$$x+\frac{105}{4}=\frac{5\sqrt{489}}{4} x+\frac{105}{4}=-\frac{5\sqrt{489}}{4}$$

Subtract \frac{105}{4} from both sides of the equation.

$$x=\frac{5\sqrt{489}-105}{4} x=\frac{-5\sqrt{489}-105}{4}$$