Solve (8+a)(900-25a)=7200+3300

$$( 8 + a ) ( 900 - 25 a ) = 7200 + 3300$$

Solve for a

$$a=6$$

$$a=22$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8+a by 900-25a and combine like terms.

$$7200+700a-25a^{2}=7200+3300$$

Add 7200 and 3300 to get 10500.

$$7200+700a-25a^{2}=10500$$

Subtract 10500 from both sides.

$$7200+700a-25a^{2}-10500=0$$

Subtract 10500 from 7200 to get -3300.

$$-3300+700a-25a^{2}=0$$

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.

$$-25a^{2}+700a-3300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, 700 for b, and -3300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-700±\sqrt{700^{2}-4\left(-25\right)\left(-3300\right)}}{2\left(-25\right)}$$

Square 700.

$$a=\frac{-700±\sqrt{490000-4\left(-25\right)\left(-3300\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$a=\frac{-700±\sqrt{490000+100\left(-3300\right)}}{2\left(-25\right)}$$

Multiply 100 times -3300.

$$a=\frac{-700±\sqrt{490000-330000}}{2\left(-25\right)}$$

Add 490000 to -330000.

$$a=\frac{-700±\sqrt{160000}}{2\left(-25\right)}$$

Take the square root of 160000.

$$a=\frac{-700±400}{2\left(-25\right)}$$

Multiply 2 times -25.

$$a=\frac{-700±400}{-50}$$

Now solve the equation a=\frac{-700±400}{-50} when ± is plus. Add -700 to 400.

$$a=-\frac{300}{-50}$$

Divide -300 by -50.

$$a=6$$

Now solve the equation a=\frac{-700±400}{-50} when ± is minus. Subtract 400 from -700.

$$a=-\frac{1100}{-50}$$

Divide -1100 by -50.

$$a=22$$

The equation is now solved.

$$a=6 a=22$$

Steps for Completing the Square

Use the distributive property to multiply 8+a by 900-25a and combine like terms.

$$7200+700a-25a^{2}=7200+3300$$

Add 7200 and 3300 to get 10500.

$$7200+700a-25a^{2}=10500$$

Subtract 7200 from both sides.

$$700a-25a^{2}=10500-7200$$

Subtract 7200 from 10500 to get 3300.

$$700a-25a^{2}=3300$$

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.

$$-25a^{2}+700a=3300$$

Divide both sides by -25.

$$\frac{-25a^{2}+700a}{-25}=\frac{3300}{-25}$$

Dividing by -25 undoes the multiplication by -25.

$$a^{2}+\frac{700}{-25}a=\frac{3300}{-25}$$

Divide 700 by -25.

$$a^{2}-28a=\frac{3300}{-25}$$

Divide 3300 by -25.

$$a^{2}-28a=-132$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-28a+\left(-14\right)^{2}=-132+\left(-14\right)^{2}$$

Square -14.

$$a^{2}-28a+196=-132+196$$

Add -132 to 196.

$$a^{2}-28a+196=64$$

Factor a^{2}-28a+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-14\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-14\right)^{2}}=\sqrt{64}$$

Simplify.

$$a-14=8 a-14=-8$$

Add 14 to both sides of the equation.

$$a=22 a=6$$