Solve x/2(28-2x)=-1800

$$\frac{ x }{ 2 } (28-2x)=-1800$$

Solve for x

$$x=-36$$

$$x=50$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(28-2x\right)=-3600$$

Use the distributive property to multiply x by 28-2x.

$$28x-2x^{2}=-3600$$

Add 3600 to both sides.

$$28x-2x^{2}+3600=0$$

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.

$$-2x^{2}+28x+3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 28 for b, and 3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-2\right)\times 3600}}{2\left(-2\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-2\right)\times 3600}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-28±\sqrt{784+8\times 3600}}{2\left(-2\right)}$$

Multiply 8 times 3600.

$$x=\frac{-28±\sqrt{784+28800}}{2\left(-2\right)}$$

Add 784 to 28800.

$$x=\frac{-28±\sqrt{29584}}{2\left(-2\right)}$$

Take the square root of 29584.

$$x=\frac{-28±172}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-28±172}{-4}$$

Now solve the equation x=\frac{-28±172}{-4} when ± is plus. Add -28 to 172.

$$x=\frac{144}{-4}$$

Divide 144 by -4.

$$x=-36$$

Now solve the equation x=\frac{-28±172}{-4} when ± is minus. Subtract 172 from -28.

$$x=-\frac{200}{-4}$$

Divide -200 by -4.

$$x=50$$

The equation is now solved.

$$x=-36 x=50$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(28-2x\right)=-3600$$

Use the distributive property to multiply x by 28-2x.

$$28x-2x^{2}=-3600$$

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.

$$-2x^{2}+28x=-3600$$

Divide both sides by -2.

$$\frac{-2x^{2}+28x}{-2}=-\frac{3600}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{28}{-2}x=-\frac{3600}{-2}$$

Divide 28 by -2.

$$x^{2}-14x=-\frac{3600}{-2}$$

Divide -3600 by -2.

$$x^{2}-14x=1800$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=1800+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=1800+49$$

Add 1800 to 49.

$$x^{2}-14x+49=1849$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=1849$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{1849}$$

Simplify.

$$x-7=43 x-7=-43$$

Add 7 to both sides of the equation.

$$x=50 x=-36$$