Solve (x-10)(920-20x)=3100

$$( x - 10 ) ( 920 - 20 x ) = 3100$$

Solve for x

$$x=15$$

$$x=41$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-10 by 920-20x and combine like terms.

$$1120x-20x^{2}-9200=3100$$

Subtract 3100 from both sides.

$$1120x-20x^{2}-9200-3100=0$$

Subtract 3100 from -9200 to get -12300.

$$1120x-20x^{2}-12300=0$$

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.

$$-20x^{2}+1120x-12300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 1120 for b, and -12300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1120±\sqrt{1120^{2}-4\left(-20\right)\left(-12300\right)}}{2\left(-20\right)}$$

Square 1120.

$$x=\frac{-1120±\sqrt{1254400-4\left(-20\right)\left(-12300\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-1120±\sqrt{1254400+80\left(-12300\right)}}{2\left(-20\right)}$$

Multiply 80 times -12300.

$$x=\frac{-1120±\sqrt{1254400-984000}}{2\left(-20\right)}$$

Add 1254400 to -984000.

$$x=\frac{-1120±\sqrt{270400}}{2\left(-20\right)}$$

Take the square root of 270400.

$$x=\frac{-1120±520}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-1120±520}{-40}$$

Now solve the equation x=\frac{-1120±520}{-40} when ± is plus. Add -1120 to 520.

$$x=-\frac{600}{-40}$$

Divide -600 by -40.

$$x=15$$

Now solve the equation x=\frac{-1120±520}{-40} when ± is minus. Subtract 520 from -1120.

$$x=-\frac{1640}{-40}$$

Divide -1640 by -40.

$$x=41$$

The equation is now solved.

$$x=15 x=41$$

Steps for Completing the Square

Use the distributive property to multiply x-10 by 920-20x and combine like terms.

$$1120x-20x^{2}-9200=3100$$

Add 9200 to both sides.

$$1120x-20x^{2}=3100+9200$$

Add 3100 and 9200 to get 12300.

$$1120x-20x^{2}=12300$$

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.

$$-20x^{2}+1120x=12300$$

Divide both sides by -20.

$$\frac{-20x^{2}+1120x}{-20}=\frac{12300}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{1120}{-20}x=\frac{12300}{-20}$$

Divide 1120 by -20.

$$x^{2}-56x=\frac{12300}{-20}$$

Divide 12300 by -20.

$$x^{2}-56x=-615$$

Divide -56, the coefficient of the x term, by 2 to get -28. Then add the square of -28 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-56x+\left(-28\right)^{2}=-615+\left(-28\right)^{2}$$

Square -28.

$$x^{2}-56x+784=-615+784$$

Add -615 to 784.

$$x^{2}-56x+784=169$$

Factor x^{2}-56x+784. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-28\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-28\right)^{2}}=\sqrt{169}$$

Simplify.

$$x-28=13 x-28=-13$$

Add 28 to both sides of the equation.

$$x=41 x=15$$