Solve (x+1)(x+1/2)=100

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

Solve for x

$$x=10\sqrt{2}-1\approx 13.142135624$$

$$x=-10\sqrt{2}-1\approx -15.142135624$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

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

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

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

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

Subtract 200 from both sides.

$$x^{2}+2x+1-200=0$$

Subtract 200 from 1 to get -199.

$$x^{2}+2x-199=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -199 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-199\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-199\right)}}{2}$$

Multiply -4 times -199.

$$x=\frac{-2±\sqrt{4+796}}{2}$$

Add 4 to 796.

$$x=\frac{-2±\sqrt{800}}{2}$$

Take the square root of 800.

$$x=\frac{-2±20\sqrt{2}}{2}$$

Now solve the equation x=\frac{-2±20\sqrt{2}}{2} when ± is plus. Add -2 to 20\sqrt{2}.

$$x=\frac{20\sqrt{2}-2}{2}$$

Divide -2+20\sqrt{2} by 2.

$$x=10\sqrt{2}-1$$

Now solve the equation x=\frac{-2±20\sqrt{2}}{2} when ± is minus. Subtract 20\sqrt{2} from -2.

$$x=\frac{-20\sqrt{2}-2}{2}$$

Divide -2-20\sqrt{2} by 2.

$$x=-10\sqrt{2}-1$$

The equation is now solved.

$$x=10\sqrt{2}-1 x=-10\sqrt{2}-1$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

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

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=10\sqrt{2} x+1=-10\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$x=10\sqrt{2}-1 x=-10\sqrt{2}-1$$