Solve (20-x)left(100+10xright)=2240

$$(20-x) \left( 100+10x \right) = 2240$$

Solve for x

$$x=4$$

$$x=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-x by 100+10x and combine like terms.

$$2000+100x-10x^{2}=2240$$

Subtract 2240 from both sides.

$$2000+100x-10x^{2}-2240=0$$

Subtract 2240 from 2000 to get -240.

$$-240+100x-10x^{2}=0$$

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.

$$-10x^{2}+100x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 100 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-10\right)\left(-240\right)}}{2\left(-10\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-10\right)\left(-240\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-100±\sqrt{10000+40\left(-240\right)}}{2\left(-10\right)}$$

Multiply 40 times -240.

$$x=\frac{-100±\sqrt{10000-9600}}{2\left(-10\right)}$$

Add 10000 to -9600.

$$x=\frac{-100±\sqrt{400}}{2\left(-10\right)}$$

Take the square root of 400.

$$x=\frac{-100±20}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-100±20}{-20}$$

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

$$x=-\frac{80}{-20}$$

Divide -80 by -20.

$$x=4$$

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

$$x=-\frac{120}{-20}$$

Divide -120 by -20.

$$x=6$$

The equation is now solved.

$$x=4 x=6$$

Steps for Completing the Square

Use the distributive property to multiply 20-x by 100+10x and combine like terms.

$$2000+100x-10x^{2}=2240$$

Subtract 2000 from both sides.

$$100x-10x^{2}=2240-2000$$

Subtract 2000 from 2240 to get 240.

$$100x-10x^{2}=240$$

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.

$$-10x^{2}+100x=240$$

Divide both sides by -10.

$$\frac{-10x^{2}+100x}{-10}=\frac{240}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{100}{-10}x=\frac{240}{-10}$$

Divide 100 by -10.

$$x^{2}-10x=\frac{240}{-10}$$

Divide 240 by -10.

$$x^{2}-10x=-24$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-24+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-24+25$$

Add -24 to 25.

$$x^{2}-10x+25=1$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-5=1 x-5=-1$$

Add 5 to both sides of the equation.

$$x=6 x=4$$