Solve (21632)/(20000)=(1+(R)/(100))^(2) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ 21632 }{ 20000 } = { \left(1+ \frac{ R }{ 100 } \right) }^{ 2 }$$

Solve for R

$R=4$
$R=-204$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Reduce the fraction $\frac{21632}{20000}$ to lowest terms by extracting and canceling out $32$.

$$\frac{676}{625}=\left(1+\frac{R}{100}\right)^{2}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(1+\frac{R}{100}\right)^{2}$.

$$\frac{676}{625}=1+2\times \frac{R}{100}+\left(\frac{R}{100}\right)^{2}$$

Cancel out $100$, the greatest common factor in $2$ and $100$.

$$\frac{676}{625}=1+\frac{R}{50}+\left(\frac{R}{100}\right)^{2}$$

To raise $\frac{R}{100}$ to a power, raise both numerator and denominator to the power and then divide.

$$\frac{676}{625}=1+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{100^{2}}{100^{2}}$.

$$\frac{676}{625}=\frac{100^{2}}{100^{2}}+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

Since $\frac{100^{2}}{100^{2}}$ and $\frac{R^{2}}{100^{2}}$ have the same denominator, add them by adding their numerators.

$$\frac{676}{625}=\frac{100^{2}+R^{2}}{100^{2}}+\frac{R}{50}$$

Combine like terms in $100^{2}+R^{2}$.

$$\frac{676}{625}=\frac{10000+R^{2}}{100^{2}}+\frac{R}{50}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $100^{2}$ and $50$ is $10000$. Multiply $\frac{R}{50}$ times $\frac{200}{200}$.

$$\frac{676}{625}=\frac{10000+R^{2}}{10000}+\frac{200R}{10000}$$

Since $\frac{10000+R^{2}}{10000}$ and $\frac{200R}{10000}$ have the same denominator, add them by adding their numerators.

$$\frac{676}{625}=\frac{10000+R^{2}+200R}{10000}$$

Divide each term of $10000+R^{2}+200R$ by $10000$ to get $1+\frac{1}{10000}R^{2}+\frac{1}{50}R$.

$$\frac{676}{625}=1+\frac{1}{10000}R^{2}+\frac{1}{50}R$$

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

$$1+\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{676}{625}$$

Subtract $\frac{676}{625}$ from both sides.

$$1+\frac{1}{10000}R^{2}+\frac{1}{50}R-\frac{676}{625}=0$$

Subtract $\frac{676}{625}$ from $1$ to get $-\frac{51}{625}$.

$$-\frac{51}{625}+\frac{1}{10000}R^{2}+\frac{1}{50}R=0$$

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.

$$\frac{1}{10000}R^{2}+\frac{1}{50}R-\frac{51}{625}=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $\frac{1}{10000}$ for $a$, $\frac{1}{50}$ for $b$, and $-\frac{51}{625}$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$R=\frac{-\frac{1}{50}±\sqrt{\left(\frac{1}{50}\right)^{2}-4\times \frac{1}{10000}\left(-\frac{51}{625}\right)}}{2\times \frac{1}{10000}}$$

Square $\frac{1}{50}$ by squaring both the numerator and the denominator of the fraction.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{1}{2500}-4\times \frac{1}{10000}\left(-\frac{51}{625}\right)}}{2\times \frac{1}{10000}}$$

Multiply $-4$ times $\frac{1}{10000}$.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{1}{2500}-\frac{1}{2500}\left(-\frac{51}{625}\right)}}{2\times \frac{1}{10000}}$$

Multiply $-\frac{1}{2500}$ times $-\frac{51}{625}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{1}{2500}+\frac{51}{1562500}}}{2\times \frac{1}{10000}}$$

Add $\frac{1}{2500}$ to $\frac{51}{1562500}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{169}{390625}}}{2\times \frac{1}{10000}}$$

Take the square root of $\frac{169}{390625}$.

$$R=\frac{-\frac{1}{50}±\frac{13}{625}}{2\times \frac{1}{10000}}$$

Multiply $2$ times $\frac{1}{10000}$.

$$R=\frac{-\frac{1}{50}±\frac{13}{625}}{\frac{1}{5000}}$$

Now solve the equation $R=\frac{-\frac{1}{50}±\frac{13}{625}}{\frac{1}{5000}}$ when $±$ is plus. Add $-\frac{1}{50}$ to $\frac{13}{625}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R=\frac{\frac{1}{1250}}{\frac{1}{5000}}$$

Divide $\frac{1}{1250}$ by $\frac{1}{5000}$ by multiplying $\frac{1}{1250}$ by the reciprocal of $\frac{1}{5000}$.

$$R=4$$

Now solve the equation $R=\frac{-\frac{1}{50}±\frac{13}{625}}{\frac{1}{5000}}$ when $±$ is minus. Subtract $\frac{13}{625}$ from $-\frac{1}{50}$ by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$R=-\frac{\frac{51}{1250}}{\frac{1}{5000}}$$

Divide $-\frac{51}{1250}$ by $\frac{1}{5000}$ by multiplying $-\frac{51}{1250}$ by the reciprocal of $\frac{1}{5000}$.

$$R=-204$$

The equation is now solved.

$$R=4$$ $$R=-204$$

Steps for Completing the Square

Reduce the fraction $\frac{21632}{20000}$ to lowest terms by extracting and canceling out $32$.

$$\frac{676}{625}=\left(1+\frac{R}{100}\right)^{2}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(1+\frac{R}{100}\right)^{2}$.

$$\frac{676}{625}=1+2\times \frac{R}{100}+\left(\frac{R}{100}\right)^{2}$$

Cancel out $100$, the greatest common factor in $2$ and $100$.

$$\frac{676}{625}=1+\frac{R}{50}+\left(\frac{R}{100}\right)^{2}$$

To raise $\frac{R}{100}$ to a power, raise both numerator and denominator to the power and then divide.

$$\frac{676}{625}=1+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{100^{2}}{100^{2}}$.

$$\frac{676}{625}=\frac{100^{2}}{100^{2}}+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

Since $\frac{100^{2}}{100^{2}}$ and $\frac{R^{2}}{100^{2}}$ have the same denominator, add them by adding their numerators.

$$\frac{676}{625}=\frac{100^{2}+R^{2}}{100^{2}}+\frac{R}{50}$$

Combine like terms in $100^{2}+R^{2}$.

$$\frac{676}{625}=\frac{10000+R^{2}}{100^{2}}+\frac{R}{50}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $100^{2}$ and $50$ is $10000$. Multiply $\frac{R}{50}$ times $\frac{200}{200}$.

$$\frac{676}{625}=\frac{10000+R^{2}}{10000}+\frac{200R}{10000}$$

Since $\frac{10000+R^{2}}{10000}$ and $\frac{200R}{10000}$ have the same denominator, add them by adding their numerators.

$$\frac{676}{625}=\frac{10000+R^{2}+200R}{10000}$$

Divide each term of $10000+R^{2}+200R$ by $10000$ to get $1+\frac{1}{10000}R^{2}+\frac{1}{50}R$.

$$\frac{676}{625}=1+\frac{1}{10000}R^{2}+\frac{1}{50}R$$

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

$$1+\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{676}{625}$$

Subtract $1$ from both sides.

$$\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{676}{625}-1$$

Subtract $1$ from $\frac{676}{625}$ to get $\frac{51}{625}$.

$$\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{51}{625}$$

Multiply both sides by $10000$.

$$\frac{\frac{1}{10000}R^{2}+\frac{1}{50}R}{\frac{1}{10000}}=\frac{\frac{51}{625}}{\frac{1}{10000}}$$

Dividing by $\frac{1}{10000}$ undoes the multiplication by $\frac{1}{10000}$.

$$R^{2}+\frac{\frac{1}{50}}{\frac{1}{10000}}R=\frac{\frac{51}{625}}{\frac{1}{10000}}$$

Divide $\frac{1}{50}$ by $\frac{1}{10000}$ by multiplying $\frac{1}{50}$ by the reciprocal of $\frac{1}{10000}$.

$$R^{2}+200R=\frac{\frac{51}{625}}{\frac{1}{10000}}$$

Divide $\frac{51}{625}$ by $\frac{1}{10000}$ by multiplying $\frac{51}{625}$ by the reciprocal of $\frac{1}{10000}$.

$$R^{2}+200R=816$$

Divide $200$, the coefficient of the $x$ term, by $2$ to get $100$. Then add the square of $100$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$R^{2}+200R+100^{2}=816+100^{2}$$

Square $100$.

$$R^{2}+200R+10000=816+10000$$

Add $816$ to $10000$.

$$R^{2}+200R+10000=10816$$

Factor $R^{2}+200R+10000$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(R+100\right)^{2}=10816$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R+100\right)^{2}}=\sqrt{10816}$$

Simplify.

$$R+100=104$$ $$R+100=-104$$

Subtract $100$ from both sides of the equation.

$$R=4$$ $$R=-204$$