Solve left(20-xright)left(-10x+700right)=2090

$$\left( 20-x \right) \left( -10x+700 \right) = 2090$$

Solve for x

$$x=\sqrt{834}+45\approx 73.879058156$$

$$x=45-\sqrt{834}\approx 16.120941844$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-x by -10x+700 and combine like terms.

$$-900x+14000+10x^{2}=2090$$

Subtract 2090 from both sides.

$$-900x+14000+10x^{2}-2090=0$$

Subtract 2090 from 14000 to get 11910.

$$-900x+11910+10x^{2}=0$$

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.

$$10x^{2}-900x+11910=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -900 for b, and 11910 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-900\right)±\sqrt{\left(-900\right)^{2}-4\times 10\times 11910}}{2\times 10}$$

Square -900.

$$x=\frac{-\left(-900\right)±\sqrt{810000-4\times 10\times 11910}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-900\right)±\sqrt{810000-40\times 11910}}{2\times 10}$$

Multiply -40 times 11910.

$$x=\frac{-\left(-900\right)±\sqrt{810000-476400}}{2\times 10}$$

Add 810000 to -476400.

$$x=\frac{-\left(-900\right)±\sqrt{333600}}{2\times 10}$$

Take the square root of 333600.

$$x=\frac{-\left(-900\right)±20\sqrt{834}}{2\times 10}$$

The opposite of -900 is 900.

$$x=\frac{900±20\sqrt{834}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{900±20\sqrt{834}}{20}$$

Now solve the equation x=\frac{900±20\sqrt{834}}{20} when ± is plus. Add 900 to 20\sqrt{834}.

$$x=\frac{20\sqrt{834}+900}{20}$$

Divide 900+20\sqrt{834} by 20.

$$x=\sqrt{834}+45$$

Now solve the equation x=\frac{900±20\sqrt{834}}{20} when ± is minus. Subtract 20\sqrt{834} from 900.

$$x=\frac{900-20\sqrt{834}}{20}$$

Divide 900-20\sqrt{834} by 20.

$$x=45-\sqrt{834}$$

The equation is now solved.

$$x=\sqrt{834}+45 x=45-\sqrt{834}$$

Steps for Completing the Square

Use the distributive property to multiply 20-x by -10x+700 and combine like terms.

$$-900x+14000+10x^{2}=2090$$

Subtract 14000 from both sides.

$$-900x+10x^{2}=2090-14000$$

Subtract 14000 from 2090 to get -11910.

$$-900x+10x^{2}=-11910$$

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.

$$10x^{2}-900x=-11910$$

Divide both sides by 10.

$$\frac{10x^{2}-900x}{10}=-\frac{11910}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{900}{10}\right)x=-\frac{11910}{10}$$

Divide -900 by 10.

$$x^{2}-90x=-\frac{11910}{10}$$

Divide -11910 by 10.

$$x^{2}-90x=-1191$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-1191+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-1191+2025$$

Add -1191 to 2025.

$$x^{2}-90x+2025=834$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=834$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{834}$$

Simplify.

$$x-45=\sqrt{834} x-45=-\sqrt{834}$$

Add 45 to both sides of the equation.

$$x=\sqrt{834}+45 x=45-\sqrt{834}$$