Solve 960=x/2(3+2x+1)

$$960= \frac{ x }{ 2 } (3+2x+1)$$

Solve for x

$$x=-32$$

$$x=30$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$1920=x\left(3+2x+1\right)$$

Add 3 and 1 to get 4.

$$1920=x\left(4+2x\right)$$

Use the distributive property to multiply x by 4+2x.

$$1920=4x+2x^{2}$$

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

$$4x+2x^{2}=1920$$

Subtract 1920 from both sides.

$$4x+2x^{2}-1920=0$$

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.

$$2x^{2}+4x-1920=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -1920 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-1920\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-1920\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-1920\right)}}{2\times 2}$$

Multiply -8 times -1920.

$$x=\frac{-4±\sqrt{16+15360}}{2\times 2}$$

Add 16 to 15360.

$$x=\frac{-4±\sqrt{15376}}{2\times 2}$$

Take the square root of 15376.

$$x=\frac{-4±124}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±124}{4}$$

Now solve the equation x=\frac{-4±124}{4} when ± is plus. Add -4 to 124.

$$x=\frac{120}{4}$$

Divide 120 by 4.

$$x=30$$

Now solve the equation x=\frac{-4±124}{4} when ± is minus. Subtract 124 from -4.

$$x=-\frac{128}{4}$$

Divide -128 by 4.

$$x=-32$$

The equation is now solved.

$$x=30 x=-32$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$1920=x\left(3+2x+1\right)$$

Add 3 and 1 to get 4.

$$1920=x\left(4+2x\right)$$

Use the distributive property to multiply x by 4+2x.

$$1920=4x+2x^{2}$$

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

$$4x+2x^{2}=1920$$

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.

$$2x^{2}+4x=1920$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{1920}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{1920}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{1920}{2}$$

Divide 1920 by 2.

$$x^{2}+2x=960$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=960+1^{2}$$

Square 1.

$$x^{2}+2x+1=960+1$$

Add 960 to 1.

$$x^{2}+2x+1=961$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=961$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{961}$$

Simplify.

$$x+1=31 x+1=-31$$

Subtract 1 from both sides of the equation.

$$x=30 x=-32$$