Solve (x-5)(45-x-5)=4(x-5)

$$( x - 5 ) ( 45 - x - 5 ) = 4 ( x - 5 )$$

Solve for x

$$x=36$$

$$x=5$$

Steps Using the Quadratic Formula

Subtract 5 from 45 to get 40.

$$\left(x-5\right)\left(40-x\right)=4\left(x-5\right)$$

Use the distributive property to multiply x-5 by 40-x and combine like terms.

$$45x-x^{2}-200=4\left(x-5\right)$$

Use the distributive property to multiply 4 by x-5.

$$45x-x^{2}-200=4x-20$$

Subtract 4x from both sides.

$$45x-x^{2}-200-4x=-20$$

Combine 45x and -4x to get 41x.

$$41x-x^{2}-200=-20$$

Add 20 to both sides.

$$41x-x^{2}-200+20=0$$

Add -200 and 20 to get -180.

$$41x-x^{2}-180=0$$

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.

$$-x^{2}+41x-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 41 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-41±\sqrt{41^{2}-4\left(-1\right)\left(-180\right)}}{2\left(-1\right)}$$

Square 41.

$$x=\frac{-41±\sqrt{1681-4\left(-1\right)\left(-180\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-41±\sqrt{1681+4\left(-180\right)}}{2\left(-1\right)}$$

Multiply 4 times -180.

$$x=\frac{-41±\sqrt{1681-720}}{2\left(-1\right)}$$

Add 1681 to -720.

$$x=\frac{-41±\sqrt{961}}{2\left(-1\right)}$$

Take the square root of 961.

$$x=\frac{-41±31}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-41±31}{-2}$$

Now solve the equation x=\frac{-41±31}{-2} when ± is plus. Add -41 to 31.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-41±31}{-2} when ± is minus. Subtract 31 from -41.

$$x=-\frac{72}{-2}$$

Divide -72 by -2.

$$x=36$$

The equation is now solved.

$$x=5 x=36$$

Steps for Completing the Square

Subtract 5 from 45 to get 40.

$$\left(x-5\right)\left(40-x\right)=4\left(x-5\right)$$

Use the distributive property to multiply x-5 by 40-x and combine like terms.

$$45x-x^{2}-200=4\left(x-5\right)$$

Use the distributive property to multiply 4 by x-5.

$$45x-x^{2}-200=4x-20$$

Subtract 4x from both sides.

$$45x-x^{2}-200-4x=-20$$

Combine 45x and -4x to get 41x.

$$41x-x^{2}-200=-20$$

Add 200 to both sides.

$$41x-x^{2}=-20+200$$

Add -20 and 200 to get 180.

$$41x-x^{2}=180$$

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.

$$-x^{2}+41x=180$$

Divide both sides by -1.

$$\frac{-x^{2}+41x}{-1}=\frac{180}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{41}{-1}x=\frac{180}{-1}$$

Divide 41 by -1.

$$x^{2}-41x=\frac{180}{-1}$$

Divide 180 by -1.

$$x^{2}-41x=-180$$

Divide -41, the coefficient of the x term, by 2 to get -\frac{41}{2}. Then add the square of -\frac{41}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-41x+\left(-\frac{41}{2}\right)^{2}=-180+\left(-\frac{41}{2}\right)^{2}$$

Square -\frac{41}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-41x+\frac{1681}{4}=-180+\frac{1681}{4}$$

Add -180 to \frac{1681}{4}.

$$x^{2}-41x+\frac{1681}{4}=\frac{961}{4}$$

Factor x^{2}-41x+\frac{1681}{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}.

$$\left(x-\frac{41}{2}\right)^{2}=\frac{961}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{41}{2}\right)^{2}}=\sqrt{\frac{961}{4}}$$

Simplify.

$$x-\frac{41}{2}=\frac{31}{2} x-\frac{41}{2}=-\frac{31}{2}$$

Add \frac{41}{2} to both sides of the equation.

$$x=36 x=5$$