Solve [(x+1)*(x+1/2)]div2=100

$$[ ( x + 1 ) \times ( \frac { x + 1 } { 2 } ) ] \div 2 = 100$$

Solve for x

$$x=19$$

$$x=-21$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$\left(x+1\right)\times \frac{x+1}{2}=100\times 2$$

Multiply both sides of the equation by 2.

$$\left(x+1\right)\left(x+1\right)=200\times 2$$

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

$$\left(x+1\right)^{2}=200\times 2$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}+2x+1=200\times 2$$

Multiply 200 and 2 to get 400.

$$x^{2}+2x+1=400$$

Subtract 400 from both sides.

$$x^{2}+2x+1-400=0$$

Subtract 400 from 1 to get -399.

$$x^{2}+2x-399=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -399 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-399\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-399\right)}}{2}$$

Multiply -4 times -399.

$$x=\frac{-2±\sqrt{4+1596}}{2}$$

Add 4 to 1596.

$$x=\frac{-2±\sqrt{1600}}{2}$$

Take the square root of 1600.

$$x=\frac{-2±40}{2}$$

Now solve the equation x=\frac{-2±40}{2} when ± is plus. Add -2 to 40.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{-2±40}{2} when ± is minus. Subtract 40 from -2.

$$x=-\frac{42}{2}$$

Divide -42 by 2.

$$x=-21$$

The equation is now solved.

$$x=19 x=-21$$

Steps for Completing the Square

Multiply both sides by 2.

$$\left(x+1\right)\times \frac{x+1}{2}=100\times 2$$

Multiply both sides of the equation by 2.

$$\left(x+1\right)\left(x+1\right)=200\times 2$$

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

$$\left(x+1\right)^{2}=200\times 2$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}+2x+1=200\times 2$$

Multiply 200 and 2 to get 400.

$$x^{2}+2x+1=400$$

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}.

$$\left(x+1\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{400}$$

Simplify.

$$x+1=20 x+1=-20$$

Subtract 1 from both sides of the equation.

$$x=19 x=-21$$