Solve (x-10)(-2x+80)=450

$$( x - 10 ) ( - 2 x + 80 ) = 450$$

Solve for x

$$x=25$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-10 by -2x+80 and combine like terms.

$$-2x^{2}+100x-800=450$$

Subtract 450 from both sides.

$$-2x^{2}+100x-800-450=0$$

Subtract 450 from -800 to get -1250.

$$-2x^{2}+100x-1250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 100 for b, and -1250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-2\right)\left(-1250\right)}}{2\left(-2\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-2\right)\left(-1250\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-100±\sqrt{10000+8\left(-1250\right)}}{2\left(-2\right)}$$

Multiply 8 times -1250.

$$x=\frac{-100±\sqrt{10000-10000}}{2\left(-2\right)}$$

Add 10000 to -10000.

$$x=\frac{-100±\sqrt{0}}{2\left(-2\right)}$$

Take the square root of 0.

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

Multiply 2 times -2.

$$x=-\frac{100}{-4}$$

Divide -100 by -4.

$$x=25$$

Steps for Completing the Square

Use the distributive property to multiply x-10 by -2x+80 and combine like terms.

$$-2x^{2}+100x-800=450$$

Add 800 to both sides.

$$-2x^{2}+100x=450+800$$

Add 450 and 800 to get 1250.

$$-2x^{2}+100x=1250$$

Divide both sides by -2.

$$\frac{-2x^{2}+100x}{-2}=\frac{1250}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{100}{-2}x=\frac{1250}{-2}$$

Divide 100 by -2.

$$x^{2}-50x=\frac{1250}{-2}$$

Divide 1250 by -2.

$$x^{2}-50x=-625$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-625+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-625+625$$

Add -625 to 625.

$$x^{2}-50x+625=0$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-25=0 x-25=0$$

Add 25 to both sides of the equation.

$$x=25 x=25$$

The equation is now solved. Solutions are the same.

$$x=25$$