Solve 4.2=(1+20%x)(1+20%x+10%)

$$4.2=(1+20 \% x)(1+20 \% x+10 \% )$$

Solve for x

$$x=5$$

$$x=-15.5$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{20}{100} to lowest terms by extracting and canceling out 20.

$$4.2=\left(1+\frac{1}{5}x\right)\left(1+\frac{20}{100}x+\frac{10}{100}\right)$$

Reduce the fraction \frac{20}{100} to lowest terms by extracting and canceling out 20.

$$4.2=\left(1+\frac{1}{5}x\right)\left(1+\frac{1}{5}x+\frac{10}{100}\right)$$

Reduce the fraction \frac{10}{100} to lowest terms by extracting and canceling out 10.

$$4.2=\left(1+\frac{1}{5}x\right)\left(1+\frac{1}{5}x+\frac{1}{10}\right)$$

Add 1 and \frac{1}{10} to get \frac{11}{10}.

$$4.2=\left(1+\frac{1}{5}x\right)\left(\frac{11}{10}+\frac{1}{5}x\right)$$

Use the distributive property to multiply 1+\frac{1}{5}x by \frac{11}{10}+\frac{1}{5}x and combine like terms.

$$4.2=\frac{11}{10}+\frac{21}{50}x+\frac{1}{25}x^{2}$$

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

$$\frac{11}{10}+\frac{21}{50}x+\frac{1}{25}x^{2}=4.2$$

Subtract 4.2 from both sides.

$$\frac{11}{10}+\frac{21}{50}x+\frac{1}{25}x^{2}-4.2=0$$

Subtract 4.2 from \frac{11}{10} to get -\frac{31}{10}.

$$-\frac{31}{10}+\frac{21}{50}x+\frac{1}{25}x^{2}=0$$

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.

$$\frac{1}{25}x^{2}+\frac{21}{50}x-\frac{31}{10}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{25} for a, \frac{21}{50} for b, and -\frac{31}{10} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{21}{50}±\sqrt{\left(\frac{21}{50}\right)^{2}-4\times \frac{1}{25}\left(-\frac{31}{10}\right)}}{2\times \frac{1}{25}}$$

Square \frac{21}{50} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{21}{50}±\sqrt{\frac{441}{2500}-4\times \frac{1}{25}\left(-\frac{31}{10}\right)}}{2\times \frac{1}{25}}$$

Multiply -4 times \frac{1}{25}.

$$x=\frac{-\frac{21}{50}±\sqrt{\frac{441}{2500}-\frac{4}{25}\left(-\frac{31}{10}\right)}}{2\times \frac{1}{25}}$$

Multiply -\frac{4}{25} times -\frac{31}{10} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{21}{50}±\sqrt{\frac{441}{2500}+\frac{62}{125}}}{2\times \frac{1}{25}}$$

Add \frac{441}{2500} to \frac{62}{125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{21}{50}±\sqrt{\frac{1681}{2500}}}{2\times \frac{1}{25}}$$

Take the square root of \frac{1681}{2500}.

$$x=\frac{-\frac{21}{50}±\frac{41}{50}}{2\times \frac{1}{25}}$$

Multiply 2 times \frac{1}{25}.

$$x=\frac{-\frac{21}{50}±\frac{41}{50}}{\frac{2}{25}}$$

Now solve the equation x=\frac{-\frac{21}{50}±\frac{41}{50}}{\frac{2}{25}} when ± is plus. Add -\frac{21}{50} to \frac{41}{50} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{2}{5}}{\frac{2}{25}}$$

Divide \frac{2}{5} by \frac{2}{25} by multiplying \frac{2}{5} by the reciprocal of \frac{2}{25}.

$$x=5$$

Now solve the equation x=\frac{-\frac{21}{50}±\frac{41}{50}}{\frac{2}{25}} when ± is minus. Subtract \frac{41}{50} from -\frac{21}{50} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{31}{25}}{\frac{2}{25}}$$

Divide -\frac{31}{25} by \frac{2}{25} by multiplying -\frac{31}{25} by the reciprocal of \frac{2}{25}.

$$x=-\frac{31}{2}$$

The equation is now solved.

$$x=5 x=-\frac{31}{2}$$

Steps for Completing the Square

Reduce the fraction \frac{20}{100} to lowest terms by extracting and canceling out 20.

$$4.2=\left(1+\frac{1}{5}x\right)\left(1+\frac{20}{100}x+\frac{10}{100}\right)$$

Reduce the fraction \frac{20}{100} to lowest terms by extracting and canceling out 20.

$$4.2=\left(1+\frac{1}{5}x\right)\left(1+\frac{1}{5}x+\frac{10}{100}\right)$$

Reduce the fraction \frac{10}{100} to lowest terms by extracting and canceling out 10.

$$4.2=\left(1+\frac{1}{5}x\right)\left(1+\frac{1}{5}x+\frac{1}{10}\right)$$

Add 1 and \frac{1}{10} to get \frac{11}{10}.

$$4.2=\left(1+\frac{1}{5}x\right)\left(\frac{11}{10}+\frac{1}{5}x\right)$$

Use the distributive property to multiply 1+\frac{1}{5}x by \frac{11}{10}+\frac{1}{5}x and combine like terms.

$$4.2=\frac{11}{10}+\frac{21}{50}x+\frac{1}{25}x^{2}$$

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

$$\frac{11}{10}+\frac{21}{50}x+\frac{1}{25}x^{2}=4.2$$

Subtract \frac{11}{10} from both sides.

$$\frac{21}{50}x+\frac{1}{25}x^{2}=4.2-\frac{11}{10}$$

Subtract \frac{11}{10} from 4.2 to get \frac{31}{10}.

$$\frac{21}{50}x+\frac{1}{25}x^{2}=\frac{31}{10}$$

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.

$$\frac{1}{25}x^{2}+\frac{21}{50}x=\frac{31}{10}$$

Multiply both sides by 25.

$$\frac{\frac{1}{25}x^{2}+\frac{21}{50}x}{\frac{1}{25}}=\frac{\frac{31}{10}}{\frac{1}{25}}$$

Dividing by \frac{1}{25} undoes the multiplication by \frac{1}{25}.

$$x^{2}+\frac{\frac{21}{50}}{\frac{1}{25}}x=\frac{\frac{31}{10}}{\frac{1}{25}}$$

Divide \frac{21}{50} by \frac{1}{25} by multiplying \frac{21}{50} by the reciprocal of \frac{1}{25}.

$$x^{2}+\frac{21}{2}x=\frac{\frac{31}{10}}{\frac{1}{25}}$$

Divide \frac{31}{10} by \frac{1}{25} by multiplying \frac{31}{10} by the reciprocal of \frac{1}{25}.

$$x^{2}+\frac{21}{2}x=\frac{155}{2}$$

Divide \frac{21}{2}, the coefficient of the x term, by 2 to get \frac{21}{4}. Then add the square of \frac{21}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{21}{2}x+\left(\frac{21}{4}\right)^{2}=\frac{155}{2}+\left(\frac{21}{4}\right)^{2}$$

Square \frac{21}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{21}{2}x+\frac{441}{16}=\frac{155}{2}+\frac{441}{16}$$

Add \frac{155}{2} to \frac{441}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{21}{2}x+\frac{441}{16}=\frac{1681}{16}$$

Factor x^{2}+\frac{21}{2}x+\frac{441}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{21}{4}\right)^{2}=\frac{1681}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{21}{4}\right)^{2}}=\sqrt{\frac{1681}{16}}$$

Simplify.

$$x+\frac{21}{4}=\frac{41}{4} x+\frac{21}{4}=-\frac{41}{4}$$

Subtract \frac{21}{4} from both sides of the equation.

$$x=5 x=-\frac{31}{2}$$