# What is a nonlinear equation

### What is a power equation?

An equation of the form \$\$ ax ^ 2 + b = c \$\$ is called a power equation.

Any natural number can be used where the exponent \$\$ 2 \$\$ is. You are only interested in the cases \$\$ x ^ 2 \$\$ and \$\$ x ^ 3 \$\$.

When solving these equations, proceed as always until you have reached \$\$ x ^ 2 = \$\$ ... or \$\$ x ^ 3 = \$\$ ... at the end of the transformation.

Taking a number with yourself is called Potentiate. \$\$ x * x = x ^ 2 \$\$ or \$\$ x * x * x = x ^ 3 \$\$

Hence the name of the equation.

The exponent is called exponent.

### Power equations with \$\$ x ^ 2 \$\$

Simple example:

\$\$ x ^ 2 = 9 \$\$

You think about which number multiplied by itself results in \$\$ 9 \$\$.

\$\$ x = 3 \$\$

However, the 3 is not the only solution, because \$\$ - 3 \$\$ is also possible.

\$\$ x = -3 \$\$

Sample:

\$\$ 3 * 3 = 9 \$\$ and \$\$ (- 3) * (- 3) = 9 \$\$.

The solution set contains two numbers.

\$\$ L = {- 3; 3} \$\$

In the course of your math career you will in this case the second root pull:

\$\$ x ^ 2 = 9 \$\$ \$\$ | sqrt (\$\$

### Power equations with \$\$ x ^ 3 \$\$

Simple example:

\$\$ x ^ 3 = 27 \$\$

You think about which number multiplied twice by itself results in \$\$ 27 \$\$.

\$\$3*3*3=27\$\$

\$\$ x = 3 \$\$

The solution set is \$\$ L = {3} \$\$.

Or is \$\$ - 3 \$\$ also possible? No, \$\$ 3 \$\$ is the only solution.

Because \$\$ (- 3) * (- 3) * (- 3) = - 27 \$\$.

You now know immediately what the solution for the equation \$\$ x ^ 3 = -27 \$\$ is.

The solution set is \$\$ L = {- 3} \$\$.

As your math career progresses, you will in this case the third root pull:

\$\$ x ^ 3 = 27 \$\$ \$\$ | root 3 () \$\$

You can recognize the 3rd root by the small three on the top left.

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### \$\$ x ^ 2 \$\$ and negative numbers

For equations of the form \$\$ x ^ 2 = -9 \$\$ you can write down that the equation no solution owns.

The solution set is empty: \$\$ L = {\$\$ \$\$} \$\$.

There is no single number that, when multiplied by itself, produces a negative number.

### Solve a complex power equation

\$\$ 3x * (x-1) + 5 + 3x = 17 \$\$ \$\$ | \$\$ Break the brackets

\$\$ 3x ^ 2-3x + 5 + 3x = 17 \$\$ \$\$ | \$\$ summarize

\$\$ 3x ^ 2 + 5 = 17 \$\$ \$\$ | -5 \$\$

\$\$ 3x ^ 2 = 12 \$\$ \$\$ |: 3 \$\$

\$\$ x ^ 2 = 4 \$\$ \$\$ | \$\$ Consider

Which number / s result when multiplied by themselves \$\$ 4 \$\$?

\$\$ x = 2 \$\$ and \$\$ x = -2 \$\$

\$\$ L = {- 2; 2} \$\$

### Equations with \$\$ x ^ 2 \$\$ and \$\$ x \$\$

\$\$ x ^ 2-2x = 3 \$\$

Divide the equation like this:

\$\$ x ^ 2-2x = 3 \$\$ \$\$ | + 2x \$\$

\$\$ x ^ 2 \$\$ stands alone on one side.

\$\$ x ^ 2 = 2x + 3 \$\$

### Solve by trying and creating a table:

Use numbers that make sense to you. \$\$ x ^ 2 \$\$ can only take the square numbers \$\$ {1, 4, 9, 16, 25, 36, 49, ...} \$\$.

Check number \$\$ x ^ 2 \$\$ \$\$ 2x + 3 \$\$
\$\$-1\$\$ \$\$1\$\$ \$\$1\$\$
\$\$0\$\$ \$\$0\$\$ \$\$3\$\$
\$\$1\$\$ \$\$1\$\$ \$\$5\$\$
\$\$2\$\$ \$\$4\$\$ \$\$7\$\$
\$\$3\$\$ \$\$9\$\$ \$\$9\$\$

The numbers \$\$ - 1 \$\$ and \$\$ 3 \$\$ are the solutions.

\$\$ L = {- 1; 3} \$\$

You will solve this task later with a formula or graphically. Now such equations always have whole numbers as a solution.

The numbers \$\$ 1 \$\$ and \$\$ - 3 \$\$ are not solutions.

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### Mathematical representation

\$\$ | x | = 2 \$\$

There are two cases that will be considered separately.

1st case: positive

\$\$ x = 2 \$\$
2nd case: negative

\$\$ x = -2 \$\$

The solution set is \$\$ L = {- 2; 2} \$\$.

### Cordial representation ☺

\$\$|\$\$\$\$|=2\$\$

There are two possibilities:

loves me OR don't love me

(positive for me) (negative for me)

1. loves me ”gives the equation \$\$=2\$\$

2. don't love me ”gives the equation \$\$=-2\$\$

The solution set \$\$ L = {- 2; 2} \$\$ means:

• Life goes with you further.
• Life goes without further.

The amount is the distance a number for \$\$ 0 \$\$.

\$\$ | 2 | = 2 \$\$ and \$\$ | -2 | = 2 \$\$

So \$\$ | 2 | = | -2 | \$\$ also applies

The same goes for variables:

\$\$ | x | = x \$\$ and \$\$ | -x | = x \$\$

So \$\$ | x | = | -x | \$\$

Equations of the type \$\$ | x | = -2 \$\$ have no solution. \$\$ L = {\$\$ \$\$} \$\$

### There is not just a \$\$ x \$\$ in the amount?

Even then, there are two cases.

Example: \$\$ | x-4 | = 9 \$\$

1st case: \$\$ x-4 = 9 \$\$ \$\$ | + 4 \$\$

\$\$ x = 13 \$\$

2nd case: \$\$ x-4 = -9 \$\$ \$\$ | + 4 \$\$

\$\$ x = -5 \$\$

1st sample:

\$\$|13–4|=9\$\$

\$\$|9|=9\$\$

\$\$9=9\$\$

2nd sample:

\$\$|-5–4|=9\$\$

\$\$|-9|=9\$\$

\$\$9=9\$\$

The solution set is \$\$ L = {13; -5} \$\$.

It must not always give a negative and a positive solution.
The equation \$\$ | \$\$\$\$ - 5 | = 1 \$\$ has the solution set \$\$ L = {4; 6} \$\$.