Solve 450=x(100-2x)

$$450=x(100-2x)$$

Solve for x

$$x=5$$

$$x=45$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 100-2x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 450 from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 100 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-100±\sqrt{10000+8\left(-450\right)}}{2\left(-2\right)}$$

Multiply 8 times -450.

$$x=\frac{-100±\sqrt{10000-3600}}{2\left(-2\right)}$$

Add 10000 to -3600.

$$x=\frac{-100±\sqrt{6400}}{2\left(-2\right)}$$

Take the square root of 6400.

$$x=\frac{-100±80}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-100±80}{-4}$$

Now solve the equation x=\frac{-100±80}{-4} when ± is plus. Add -100 to 80.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-100±80}{-4} when ± is minus. Subtract 80 from -100.

$$x=-\frac{180}{-4}$$

Divide -180 by -4.

$$x=45$$

The equation is now solved.

$$x=5 x=45$$

Steps for Completing the Square

Use the distributive property to multiply x by 100-2x.

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

Swap sides so that all variable terms are on the left hand side.

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

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.

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

Divide both sides by -2.

$$\frac{-2x^{2}+100x}{-2}=\frac{450}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{100}{-2}x=\frac{450}{-2}$$

Divide 100 by -2.

$$x^{2}-50x=\frac{450}{-2}$$

Divide 450 by -2.

$$x^{2}-50x=-225$$

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.

$$x^{2}-50x+\left(-25\right)^{2}=-225+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-225+625$$

Add -225 to 625.

$$x^{2}-50x+625=400$$

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}.

$$\left(x-25\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-25=20 x-25=-20$$

Add 25 to both sides of the equation.

$$x=45 x=5$$