Solve 9=2x(-5-x/2)

$$9=2x(-5- \frac{ x }{ 2 } )$$

Solve for x

$$x=-9$$

$$x=-1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$18=4x\left(-5-\frac{x}{2}\right)$$

Use the distributive property to multiply 4x by -5-\frac{x}{2}.

$$18=-20x+4x\left(-\frac{x}{2}\right)$$

Cancel out 2, the greatest common factor in 4 and 2.

$$18=-20x-2xx$$

Multiply x and x to get x^{2}.

$$18=-20x-2x^{2}$$

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

$$-20x-2x^{2}=18$$

Subtract 18 from both sides.

$$-20x-2x^{2}-18=0$$

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.

$$-2x^{2}-20x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -20 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-2\right)\left(-18\right)}}{2\left(-2\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-2\right)\left(-18\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-20\right)±\sqrt{400+8\left(-18\right)}}{2\left(-2\right)}$$

Multiply 8 times -18.

$$x=\frac{-\left(-20\right)±\sqrt{400-144}}{2\left(-2\right)}$$

Add 400 to -144.

$$x=\frac{-\left(-20\right)±\sqrt{256}}{2\left(-2\right)}$$

Take the square root of 256.

$$x=\frac{-\left(-20\right)±16}{2\left(-2\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±16}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{20±16}{-4}$$

Now solve the equation x=\frac{20±16}{-4} when ± is plus. Add 20 to 16.

$$x=\frac{36}{-4}$$

Divide 36 by -4.

$$x=-9$$

Now solve the equation x=\frac{20±16}{-4} when ± is minus. Subtract 16 from 20.

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

Divide 4 by -4.

$$x=-1$$

The equation is now solved.

$$x=-9 x=-1$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$18=4x\left(-5-\frac{x}{2}\right)$$

Use the distributive property to multiply 4x by -5-\frac{x}{2}.

$$18=-20x+4x\left(-\frac{x}{2}\right)$$

Cancel out 2, the greatest common factor in 4 and 2.

$$18=-20x-2xx$$

Multiply x and x to get x^{2}.

$$18=-20x-2x^{2}$$

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

$$-20x-2x^{2}=18$$

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.

$$-2x^{2}-20x=18$$

Divide both sides by -2.

$$\frac{-2x^{2}-20x}{-2}=\frac{18}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{20}{-2}\right)x=\frac{18}{-2}$$

Divide -20 by -2.

$$x^{2}+10x=\frac{18}{-2}$$

Divide 18 by -2.

$$x^{2}+10x=-9$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=-9+5^{2}$$

Square 5.

$$x^{2}+10x+25=-9+25$$

Add -9 to 25.

$$x^{2}+10x+25=16$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+5=4 x+5=-4$$

Subtract 5 from both sides of the equation.

$$x=-1 x=-9$$