Solve (22-x)(17-x)=300

$$( 22 - x ) ( 17 - x ) = 300$$

Solve for x

$$x=37$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 22-x by 17-x and combine like terms.

$$374-39x+x^{2}=300$$

Subtract 300 from both sides.

$$374-39x+x^{2}-300=0$$

Subtract 300 from 374 to get 74.

$$74-39x+x^{2}=0$$

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.

$$x^{2}-39x+74=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -39 for b, and 74 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-39\right)±\sqrt{\left(-39\right)^{2}-4\times 74}}{2}$$

Square -39.

$$x=\frac{-\left(-39\right)±\sqrt{1521-4\times 74}}{2}$$

Multiply -4 times 74.

$$x=\frac{-\left(-39\right)±\sqrt{1521-296}}{2}$$

Add 1521 to -296.

$$x=\frac{-\left(-39\right)±\sqrt{1225}}{2}$$

Take the square root of 1225.

$$x=\frac{-\left(-39\right)±35}{2}$$

The opposite of -39 is 39.

$$x=\frac{39±35}{2}$$

Now solve the equation x=\frac{39±35}{2} when ± is plus. Add 39 to 35.

$$x=\frac{74}{2}$$

Divide 74 by 2.

$$x=37$$

Now solve the equation x=\frac{39±35}{2} when ± is minus. Subtract 35 from 39.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=37 x=2$$

Steps for Completing the Square

Use the distributive property to multiply 22-x by 17-x and combine like terms.

$$374-39x+x^{2}=300$$

Subtract 374 from both sides.

$$-39x+x^{2}=300-374$$

Subtract 374 from 300 to get -74.

$$-39x+x^{2}=-74$$

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.

$$x^{2}-39x=-74$$

Divide -39, the coefficient of the x term, by 2 to get -\frac{39}{2}. Then add the square of -\frac{39}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-39x+\left(-\frac{39}{2}\right)^{2}=-74+\left(-\frac{39}{2}\right)^{2}$$

Square -\frac{39}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-39x+\frac{1521}{4}=-74+\frac{1521}{4}$$

Add -74 to \frac{1521}{4}.

$$x^{2}-39x+\frac{1521}{4}=\frac{1225}{4}$$

Factor x^{2}-39x+\frac{1521}{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}.

$$\left(x-\frac{39}{2}\right)^{2}=\frac{1225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Simplify.

$$x-\frac{39}{2}=\frac{35}{2} x-\frac{39}{2}=-\frac{35}{2}$$

Add \frac{39}{2} to both sides of the equation.

$$x=37 x=2$$