Solve (1 + i) ^ 100 = 2 ^ 49 * (x + iy) అయితే x ^ 2 + y ^ 2 విలువ ఎంత ? | 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

$$( 1 + i ) ^ { 100 } = 2 ^ { 49 } ( x + i y ) , 600000 ^ { - 1 } x ^ { 2 } + y ^ { 2 } = 900000 ?$$

Solve for x, y

$x=\frac{-1200000+400\sqrt{2024996624985}i}{599999}\approx -2.000003333+948.684088617i\text{, }y=-\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\approx -948.684088617-0.000003333i$
$x=\frac{-400\sqrt{2024996624985}i-1200000}{599999}\approx -2.000003333-948.684088617i\text{, }y=\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\approx 948.684088617-0.000003333i$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Calculate $1+i$ to the power of $100$ and get $-1125899906842624$.

$$-1125899906842624=2^{49}\left(x+iy\right)$$

Calculate $2$ to the power of $49$ and get $562949953421312$.

$$-1125899906842624=562949953421312\left(x+iy\right)$$

Use the distributive property to multiply $562949953421312$ by $x+iy$.

$$-1125899906842624=562949953421312x+562949953421312iy$$

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

$$562949953421312x+562949953421312iy=-1125899906842624$$

Consider the second equation. Calculate $600000$ to the power of $-1$ and get $\frac{1}{600000}$.

$$\frac{1}{600000}x^{2}+y^{2}=900000$$

Solve $562949953421312x+562949953421312iy=-1125899906842624$ for $x$ by isolating $x$ on the left hand side of the equal sign.

$$562949953421312x+562949953421312iy=-1125899906842624$$

Subtract $562949953421312iy$ from both sides of the equation.

$$562949953421312x=-562949953421312iy-1125899906842624$$

Divide both sides by $562949953421312$.

$$x=-iy-2$$

Substitute $-iy-2$ for $x$ in the other equation, $y^{2}+\frac{1}{600000}x^{2}=900000$.

$$y^{2}+\frac{1}{600000}\left(-iy-2\right)^{2}=900000$$

Square $-iy-2$.

$$y^{2}+\frac{1}{600000}\left(-y^{2}+4iy+4\right)=900000$$

Multiply $\frac{1}{600000}$ times $-y^{2}+4iy+4$.

$$y^{2}-\frac{1}{600000}y^{2}+\frac{1}{150000}iy+\frac{1}{150000}=900000$$

Add $y^{2}$ to $-\frac{1}{600000}y^{2}$.

$$\frac{599999}{600000}y^{2}+\frac{1}{150000}iy+\frac{1}{150000}=900000$$

Subtract $900000$ from both sides of the equation.

$$\frac{599999}{600000}y^{2}+\frac{1}{150000}iy-\frac{134999999999}{150000}=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $1+\frac{1}{600000}\left(-i\right)^{2}$ for $a$, $\frac{1}{600000}\left(-2\right)\left(-i\right)\times 2$ for $b$, and $-\frac{134999999999}{150000}$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$y=\frac{-\frac{1}{150000}i±\sqrt{\left(\frac{1}{150000}i\right)^{2}-4\times \frac{599999}{600000}\left(-\frac{134999999999}{150000}\right)}}{2\times \frac{599999}{600000}}$$

Square $\frac{1}{600000}\left(-2\right)\left(-i\right)\times 2$.

$$y=\frac{-\frac{1}{150000}i±\sqrt{-\frac{1}{22500000000}-4\times \frac{599999}{600000}\left(-\frac{134999999999}{150000}\right)}}{2\times \frac{599999}{600000}}$$

Multiply $-4$ times $1+\frac{1}{600000}\left(-i\right)^{2}$.

$$y=\frac{-\frac{1}{150000}i±\sqrt{-\frac{1}{22500000000}-\frac{599999}{150000}\left(-\frac{134999999999}{150000}\right)}}{2\times \frac{599999}{600000}}$$

Multiply $-\frac{599999}{150000}$ times $-\frac{134999999999}{150000}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{1}{150000}i±\sqrt{\frac{-1+80999864999400001}{22500000000}}}{2\times \frac{599999}{600000}}$$

Add $-\frac{1}{22500000000}$ to $\frac{80999864999400001}{22500000000}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{1}{150000}i±\sqrt{\frac{134999774999}{37500}}}{2\times \frac{599999}{600000}}$$

Take the square root of $\frac{134999774999}{37500}$.

$$y=\frac{-\frac{1}{150000}i±\frac{\sqrt{2024996624985}}{750}}{2\times \frac{599999}{600000}}$$

Multiply $2$ times $1+\frac{1}{600000}\left(-i\right)^{2}$.

$$y=\frac{-\frac{1}{150000}i±\frac{\sqrt{2024996624985}}{750}}{\frac{599999}{300000}}$$

Now solve the equation $y=\frac{-\frac{1}{150000}i±\frac{\sqrt{2024996624985}}{750}}{\frac{599999}{300000}}$ when $±$ is plus. Add $-\frac{1}{150000}i$ to $\frac{\sqrt{2024996624985}}{750}$.

$$y=\frac{\frac{\sqrt{2024996624985}}{750}-\frac{1}{150000}i}{\frac{599999}{300000}}$$

Divide $-\frac{1}{150000}i+\frac{\sqrt{2024996624985}}{750}$ by $\frac{599999}{300000}$ by multiplying $-\frac{1}{150000}i+\frac{\sqrt{2024996624985}}{750}$ by the reciprocal of $\frac{599999}{300000}$.

$$y=\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i$$

Now solve the equation $y=\frac{-\frac{1}{150000}i±\frac{\sqrt{2024996624985}}{750}}{\frac{599999}{300000}}$ when $±$ is minus. Subtract $\frac{\sqrt{2024996624985}}{750}$ from $-\frac{1}{150000}i$.

$$y=\frac{-\frac{\sqrt{2024996624985}}{750}-\frac{1}{150000}i}{\frac{599999}{300000}}$$

Divide $-\frac{1}{150000}i-\frac{\sqrt{2024996624985}}{750}$ by $\frac{599999}{300000}$ by multiplying $-\frac{1}{150000}i-\frac{\sqrt{2024996624985}}{750}$ by the reciprocal of $\frac{599999}{300000}$.

$$y=-\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i$$

There are two solutions for $y$: $-\frac{2}{599999}i+\frac{400\sqrt{2024996624985}}{599999}$ and $-\frac{2}{599999}i-\frac{400\sqrt{2024996624985}}{599999}$. Substitute $-\frac{2}{599999}i+\frac{400\sqrt{2024996624985}}{599999}$ for $y$ in the equation $x=-iy-2$ to find the corresponding solution for $x$ that satisfies both equations.

$$x=-i\left(\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\right)-2$$

Now substitute $-\frac{2}{599999}i-\frac{400\sqrt{2024996624985}}{599999}$ for $y$ in the equation $x=-iy-2$ and solve to find the corresponding solution for $x$ that satisfies both equations.

$$x=-i\left(-\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\right)-2$$

The system is now solved.

$$x=-i\left(\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\right)-2,y=\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\text{ or }x=-i\left(-\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i\right)-2,y=-\frac{400\sqrt{2024996624985}}{599999}-\frac{2}{599999}i$$