Solve (a-2)(350-10a)=400

$$( a - 2 ) ( 350 - 10 a ) = 400$$

Solve for a

$$a = \frac{\sqrt{929} + 37}{2} \approx 33.739750654$$

$$a = \frac{37 - \sqrt{929}}{2} \approx 3.260249346$$

Steps Using the Quadratic Formula

Use the distributive property to multiply a-2 by 350-10a and combine like terms.

$$370a-10a^{2}-700=400$$

Subtract 400 from both sides.

$$370a-10a^{2}-700-400=0$$

Subtract 400 from -700 to get -1100.

$$370a-10a^{2}-1100=0$$

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.

$$-10a^{2}+370a-1100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 370 for b, and -1100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-370±\sqrt{370^{2}-4\left(-10\right)\left(-1100\right)}}{2\left(-10\right)}$$

Square 370.

$$a=\frac{-370±\sqrt{136900-4\left(-10\right)\left(-1100\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$a=\frac{-370±\sqrt{136900+40\left(-1100\right)}}{2\left(-10\right)}$$

Multiply 40 times -1100.

$$a=\frac{-370±\sqrt{136900-44000}}{2\left(-10\right)}$$

Add 136900 to -44000.

$$a=\frac{-370±\sqrt{92900}}{2\left(-10\right)}$$

Take the square root of 92900.

$$a=\frac{-370±10\sqrt{929}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$a=\frac{-370±10\sqrt{929}}{-20}$$

Now solve the equation a=\frac{-370±10\sqrt{929}}{-20} when ± is plus. Add -370 to 10\sqrt{929}.

$$a=\frac{10\sqrt{929}-370}{-20}$$

Divide -370+10\sqrt{929} by -20.

$$a=\frac{37-\sqrt{929}}{2}$$

Now solve the equation a=\frac{-370±10\sqrt{929}}{-20} when ± is minus. Subtract 10\sqrt{929} from -370.

$$a=\frac{-10\sqrt{929}-370}{-20}$$

Divide -370-10\sqrt{929} by -20.

$$a=\frac{\sqrt{929}+37}{2}$$

The equation is now solved.

$$a=\frac{37-\sqrt{929}}{2} a=\frac{\sqrt{929}+37}{2}$$

Steps for Completing the Square

Use the distributive property to multiply a-2 by 350-10a and combine like terms.

$$370a-10a^{2}-700=400$$

Add 700 to both sides.

$$370a-10a^{2}=400+700$$

Add 400 and 700 to get 1100.

$$370a-10a^{2}=1100$$

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.

$$-10a^{2}+370a=1100$$

Divide both sides by -10.

$$\frac{-10a^{2}+370a}{-10}=\frac{1100}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$a^{2}+\frac{370}{-10}a=\frac{1100}{-10}$$

Divide 370 by -10.

$$a^{2}-37a=\frac{1100}{-10}$$

Divide 1100 by -10.

$$a^{2}-37a=-110$$

Divide -37, the coefficient of the x term, by 2 to get -\frac{37}{2}. Then add the square of -\frac{37}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-37a+\left(-\frac{37}{2}\right)^{2}=-110+\left(-\frac{37}{2}\right)^{2}$$

Square -\frac{37}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-37a+\frac{1369}{4}=-110+\frac{1369}{4}$$

Add -110 to \frac{1369}{4}.

$$a^{2}-37a+\frac{1369}{4}=\frac{929}{4}$$

Factor a^{2}-37a+\frac{1369}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{37}{2}\right)^{2}=\frac{929}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{37}{2}\right)^{2}}=\sqrt{\frac{929}{4}}$$

Simplify.

$$a-\frac{37}{2}=\frac{\sqrt{929}}{2} a-\frac{37}{2}=-\frac{\sqrt{929}}{2}$$

Add \frac{37}{2} to both sides of the equation.

$$a=\frac{\sqrt{929}+37}{2} a=\frac{37-\sqrt{929}}{2}$$