Solve (100-2x)(x-30)=200

$$(100-2x)(x-30)=200$$

Solve for x

$$x=40$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-2x by x-30 and combine like terms.

$$160x-3000-2x^{2}=200$$

Subtract 200 from both sides.

$$160x-3000-2x^{2}-200=0$$

Subtract 200 from -3000 to get -3200.

$$160x-3200-2x^{2}=0$$

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.

$$-2x^{2}+160x-3200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 160 for b, and -3200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\left(-2\right)\left(-3200\right)}}{2\left(-2\right)}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\left(-2\right)\left(-3200\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-160±\sqrt{25600+8\left(-3200\right)}}{2\left(-2\right)}$$

Multiply 8 times -3200.

$$x=\frac{-160±\sqrt{25600-25600}}{2\left(-2\right)}$$

Add 25600 to -25600.

$$x=\frac{-160±\sqrt{0}}{2\left(-2\right)}$$

Take the square root of 0.

$$x=-\frac{160}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=-\frac{160}{-4}$$

Divide -160 by -4.

$$x=40$$

Steps for Completing the Square

Use the distributive property to multiply 100-2x by x-30 and combine like terms.

$$160x-3000-2x^{2}=200$$

Add 3000 to both sides.

$$160x-2x^{2}=200+3000$$

Add 200 and 3000 to get 3200.

$$160x-2x^{2}=3200$$

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.

$$-2x^{2}+160x=3200$$

Divide both sides by -2.

$$\frac{-2x^{2}+160x}{-2}=\frac{3200}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{160}{-2}x=\frac{3200}{-2}$$

Divide 160 by -2.

$$x^{2}-80x=\frac{3200}{-2}$$

Divide 3200 by -2.

$$x^{2}-80x=-1600$$

Divide -80, the coefficient of the x term, by 2 to get -40. Then add the square of -40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-80x+\left(-40\right)^{2}=-1600+\left(-40\right)^{2}$$

Square -40.

$$x^{2}-80x+1600=-1600+1600$$

Add -1600 to 1600.

$$x^{2}-80x+1600=0$$

Factor x^{2}-80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-40\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-40=0 x-40=0$$

Add 40 to both sides of the equation.

$$x=40 x=40$$

The equation is now solved. Solutions are the same.

$$x=40$$