Solve (44-x)(20+5x)=1600

$$( 44 - x ) ( 20 + 5 x ) = 1600$$

Solve for x

$$x=4$$

$$x=36$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 44-x by 20+5x and combine like terms.

$$880+200x-5x^{2}=1600$$

Subtract 1600 from both sides.

$$880+200x-5x^{2}-1600=0$$

Subtract 1600 from 880 to get -720.

$$-720+200x-5x^{2}=0$$

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.

$$-5x^{2}+200x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 200 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-5\right)\left(-720\right)}}{2\left(-5\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-5\right)\left(-720\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-200±\sqrt{40000+20\left(-720\right)}}{2\left(-5\right)}$$

Multiply 20 times -720.

$$x=\frac{-200±\sqrt{40000-14400}}{2\left(-5\right)}$$

Add 40000 to -14400.

$$x=\frac{-200±\sqrt{25600}}{2\left(-5\right)}$$

Take the square root of 25600.

$$x=\frac{-200±160}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-200±160}{-10}$$

Now solve the equation x=\frac{-200±160}{-10} when ± is plus. Add -200 to 160.

$$x=-\frac{40}{-10}$$

Divide -40 by -10.

$$x=4$$

Now solve the equation x=\frac{-200±160}{-10} when ± is minus. Subtract 160 from -200.

$$x=-\frac{360}{-10}$$

Divide -360 by -10.

$$x=36$$

The equation is now solved.

$$x=4 x=36$$

Steps for Completing the Square

Use the distributive property to multiply 44-x by 20+5x and combine like terms.

$$880+200x-5x^{2}=1600$$

Subtract 880 from both sides.

$$200x-5x^{2}=1600-880$$

Subtract 880 from 1600 to get 720.

$$200x-5x^{2}=720$$

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.

$$-5x^{2}+200x=720$$

Divide both sides by -5.

$$\frac{-5x^{2}+200x}{-5}=\frac{720}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{200}{-5}x=\frac{720}{-5}$$

Divide 200 by -5.

$$x^{2}-40x=\frac{720}{-5}$$

Divide 720 by -5.

$$x^{2}-40x=-144$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-144+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-144+400$$

Add -144 to 400.

$$x^{2}-40x+400=256$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=256$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{256}$$

Simplify.

$$x-20=16 x-20=-16$$

Add 20 to both sides of the equation.

$$x=36 x=4$$