Solve (2x-4)(x-4)=(5-x)(4-x)

$$( 2 x - 4 ) ( x - 4 ) = ( 5 - x ) ( 4 - x )$$

Solve for x

$$x=-1$$

$$x=4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-4 by x-4 and combine like terms.

$$2x^{2}-12x+16=\left(5-x\right)\left(4-x\right)$$

Use the distributive property to multiply 5-x by 4-x and combine like terms.

$$2x^{2}-12x+16=20-9x+x^{2}$$

Subtract 20 from both sides.

$$2x^{2}-12x+16-20=-9x+x^{2}$$

Subtract 20 from 16 to get -4.

$$2x^{2}-12x-4=-9x+x^{2}$$

Add 9x to both sides.

$$2x^{2}-12x-4+9x=x^{2}$$

Combine -12x and 9x to get -3x.

$$2x^{2}-3x-4=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-3x-4-x^{2}=0$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-3x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-4\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$x=\frac{-\left(-3\right)±\sqrt{9+16}}{2}$$

Add 9 to 16.

$$x=\frac{-\left(-3\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-3\right)±5}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±5}{2}$$

Now solve the equation x=\frac{3±5}{2} when ± is plus. Add 3 to 5.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{3±5}{2} when ± is minus. Subtract 5 from 3.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=4 x=-1$$

Steps for Completing the Square

Use the distributive property to multiply 2x-4 by x-4 and combine like terms.

$$2x^{2}-12x+16=\left(5-x\right)\left(4-x\right)$$

Use the distributive property to multiply 5-x by 4-x and combine like terms.

$$2x^{2}-12x+16=20-9x+x^{2}$$

Add 9x to both sides.

$$2x^{2}-12x+16+9x=20+x^{2}$$

Combine -12x and 9x to get -3x.

$$2x^{2}-3x+16=20+x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-3x+16-x^{2}=20$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-3x+16=20$$

Subtract 16 from both sides.

$$x^{2}-3x=20-16$$

Subtract 16 from 20 to get 4.

$$x^{2}-3x=4$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=4+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=4+\frac{9}{4}$$

Add 4 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{25}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{5}{2} x-\frac{3}{2}=-\frac{5}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=4 x=-1$$