Solve ((20-a)(380+5a)=32000

$$( ( 20 - a ) ( 380 + 5 a ) = 32000$$

Solve for a

$$a=-28+64i$$

$$a=-28-64i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-a by 380+5a and combine like terms.

$$7600-280a-5a^{2}=32000$$

Subtract 32000 from both sides.

$$7600-280a-5a^{2}-32000=0$$

Subtract 32000 from 7600 to get -24400.

$$-24400-280a-5a^{2}=0$$

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.

$$-5a^{2}-280a-24400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -280 for b, and -24400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-280\right)±\sqrt{\left(-280\right)^{2}-4\left(-5\right)\left(-24400\right)}}{2\left(-5\right)}$$

Square -280.

$$a=\frac{-\left(-280\right)±\sqrt{78400-4\left(-5\right)\left(-24400\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$a=\frac{-\left(-280\right)±\sqrt{78400+20\left(-24400\right)}}{2\left(-5\right)}$$

Multiply 20 times -24400.

$$a=\frac{-\left(-280\right)±\sqrt{78400-488000}}{2\left(-5\right)}$$

Add 78400 to -488000.

$$a=\frac{-\left(-280\right)±\sqrt{-409600}}{2\left(-5\right)}$$

Take the square root of -409600.

$$a=\frac{-\left(-280\right)±640i}{2\left(-5\right)}$$

The opposite of -280 is 280.

$$a=\frac{280±640i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$a=\frac{280±640i}{-10}$$

Now solve the equation a=\frac{280±640i}{-10} when ± is plus. Add 280 to 640i.

$$a=\frac{280+640i}{-10}$$

Divide 280+640i by -10.

$$a=-28-64i$$

Now solve the equation a=\frac{280±640i}{-10} when ± is minus. Subtract 640i from 280.

$$a=\frac{280-640i}{-10}$$

Divide 280-640i by -10.

$$a=-28+64i$$

The equation is now solved.

$$a=-28-64i a=-28+64i$$

Steps for Completing the Square

Use the distributive property to multiply 20-a by 380+5a and combine like terms.

$$7600-280a-5a^{2}=32000$$

Subtract 7600 from both sides.

$$-280a-5a^{2}=32000-7600$$

Subtract 7600 from 32000 to get 24400.

$$-280a-5a^{2}=24400$$

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.

$$-5a^{2}-280a=24400$$

Divide both sides by -5.

$$\frac{-5a^{2}-280a}{-5}=\frac{24400}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$a^{2}+\left(-\frac{280}{-5}\right)a=\frac{24400}{-5}$$

Divide -280 by -5.

$$a^{2}+56a=\frac{24400}{-5}$$

Divide 24400 by -5.

$$a^{2}+56a=-4880$$

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.

$$a^{2}+56a+28^{2}=-4880+28^{2}$$

Square 28.

$$a^{2}+56a+784=-4880+784$$

Add -4880 to 784.

$$a^{2}+56a+784=-4096$$

Factor a^{2}+56a+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}.

$$\left(a+28\right)^{2}=-4096$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+28\right)^{2}}=\sqrt{-4096}$$

Simplify.

$$a+28=64i a+28=-64i$$

Subtract 28 from both sides of the equation.

$$a=-28+64i a=-28-64i$$