Solve 54(1+x)(1+x)=121.5

$$54(1+x)(1+x)=121.5$$

Solve for x

$$x=0.5$$

$$x=-2.5$$

Steps Using the Quadratic Formula

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

$$54\left(1+x\right)^{2}=121.5$$

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

$$54\left(1+2x+x^{2}\right)=121.5$$

Use the distributive property to multiply 54 by 1+2x+x^{2}.

$$54+108x+54x^{2}=121.5$$

Subtract 121.5 from both sides.

$$54+108x+54x^{2}-121.5=0$$

Subtract 121.5 from 54 to get -67.5.

$$-67.5+108x+54x^{2}=0$$

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.

$$54x^{2}+108x-67.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 54 for a, 108 for b, and -67.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}-4\times 54\left(-67.5\right)}}{2\times 54}$$

Square 108.

$$x=\frac{-108±\sqrt{11664-4\times 54\left(-67.5\right)}}{2\times 54}$$

Multiply -4 times 54.

$$x=\frac{-108±\sqrt{11664-216\left(-67.5\right)}}{2\times 54}$$

Multiply -216 times -67.5.

$$x=\frac{-108±\sqrt{11664+14580}}{2\times 54}$$

Add 11664 to 14580.

$$x=\frac{-108±\sqrt{26244}}{2\times 54}$$

Take the square root of 26244.

$$x=\frac{-108±162}{2\times 54}$$

Multiply 2 times 54.

$$x=\frac{-108±162}{108}$$

Now solve the equation x=\frac{-108±162}{108} when ± is plus. Add -108 to 162.

$$x=\frac{54}{108}$$

Reduce the fraction \frac{54}{108} to lowest terms by extracting and canceling out 54.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{-108±162}{108} when ± is minus. Subtract 162 from -108.

$$x=-\frac{270}{108}$$

Reduce the fraction \frac{-270}{108} to lowest terms by extracting and canceling out 54.

$$x=-\frac{5}{2}$$

The equation is now solved.

$$x=\frac{1}{2} x=-\frac{5}{2}$$

Steps for Completing the Square

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

$$54\left(1+x\right)^{2}=121.5$$

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

$$54\left(1+2x+x^{2}\right)=121.5$$

Use the distributive property to multiply 54 by 1+2x+x^{2}.

$$54+108x+54x^{2}=121.5$$

Subtract 54 from both sides.

$$108x+54x^{2}=121.5-54$$

Subtract 54 from 121.5 to get 67.5.

$$108x+54x^{2}=67.5$$

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.

$$54x^{2}+108x=67.5$$

Divide both sides by 54.

$$\frac{54x^{2}+108x}{54}=\frac{67.5}{54}$$

Dividing by 54 undoes the multiplication by 54.

$$x^{2}+\frac{108}{54}x=\frac{67.5}{54}$$

Divide 108 by 54.

$$x^{2}+2x=\frac{67.5}{54}$$

Divide 67.5 by 54.

$$x^{2}+2x=1.25$$

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.

$$x^{2}+2x+1^{2}=1.25+1^{2}$$

Square 1.

$$x^{2}+2x+1=1.25+1$$

Add 1.25 to 1.

$$x^{2}+2x+1=2.25$$

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}.

$$\left(x+1\right)^{2}=2.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{2.25}$$

Simplify.

$$x+1=\frac{3}{2} x+1=-\frac{3}{2}$$

Subtract 1 from both sides of the equation.

$$x=\frac{1}{2} x=-\frac{5}{2}$$