Solve (4x-8)(x+5)=(5x-2)(x-2)

$$( 4 x - 8 ) ( x + 5 ) = ( 5 x - 2 ) ( x - 2 )$$

Solve for x

$$x=22$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x-8 by x+5 and combine like terms.

$$4x^{2}+12x-40=\left(5x-2\right)\left(x-2\right)$$

Use the distributive property to multiply 5x-2 by x-2 and combine like terms.

$$4x^{2}+12x-40=5x^{2}-12x+4$$

Subtract 5x^{2} from both sides.

$$4x^{2}+12x-40-5x^{2}=-12x+4$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+12x-40=-12x+4$$

Add 12x to both sides.

$$-x^{2}+12x-40+12x=4$$

Combine 12x and 12x to get 24x.

$$-x^{2}+24x-40=4$$

Subtract 4 from both sides.

$$-x^{2}+24x-40-4=0$$

Subtract 4 from -40 to get -44.

$$-x^{2}+24x-44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and -44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-44\right)}}{2\left(-1\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-1\right)\left(-44\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-24±\sqrt{576+4\left(-44\right)}}{2\left(-1\right)}$$

Multiply 4 times -44.

$$x=\frac{-24±\sqrt{576-176}}{2\left(-1\right)}$$

Add 576 to -176.

$$x=\frac{-24±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$x=\frac{-24±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±20}{-2}$$

Now solve the equation x=\frac{-24±20}{-2} when ± is plus. Add -24 to 20.

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

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-24±20}{-2} when ± is minus. Subtract 20 from -24.

$$x=-\frac{44}{-2}$$

Divide -44 by -2.

$$x=22$$

The equation is now solved.

$$x=2 x=22$$

Steps for Completing the Square

Use the distributive property to multiply 4x-8 by x+5 and combine like terms.

$$4x^{2}+12x-40=\left(5x-2\right)\left(x-2\right)$$

Use the distributive property to multiply 5x-2 by x-2 and combine like terms.

$$4x^{2}+12x-40=5x^{2}-12x+4$$

Subtract 5x^{2} from both sides.

$$4x^{2}+12x-40-5x^{2}=-12x+4$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+12x-40=-12x+4$$

Add 12x to both sides.

$$-x^{2}+12x-40+12x=4$$

Combine 12x and 12x to get 24x.

$$-x^{2}+24x-40=4$$

Add 40 to both sides.

$$-x^{2}+24x=4+40$$

Add 4 and 40 to get 44.

$$-x^{2}+24x=44$$

Divide both sides by -1.

$$\frac{-x^{2}+24x}{-1}=\frac{44}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{24}{-1}x=\frac{44}{-1}$$

Divide 24 by -1.

$$x^{2}-24x=\frac{44}{-1}$$

Divide 44 by -1.

$$x^{2}-24x=-44$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=-44+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-44+144$$

Add -44 to 144.

$$x^{2}-24x+144=100$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-12=10 x-12=-10$$

Add 12 to both sides of the equation.

$$x=22 x=2$$