# What is x if 4x 5x 40

### What is a quadratic equation?

In a quadratic equation, the variable occurs to the power of two and not higher.

#### Examples

- $$ x ^ 2 = 3 $$
- $$ 2x ^ 2 + 1.5x = 0 $$
- $$ x ^ 2 + 2x - 3 = 0 $$
- $$ 0.5x ^ 2 - 3x = 1.5 $$

In addition to the quadratic term ($$ x ^ 2 $$), quadratic equations can contain a linear ($$ x $$) and an absolute term (a number).

#### example

$$ 0.5 x ^ 2 $$ (quadratic term) $$ - 3 x $$ (linear term) = $$ 1.5 $$ (absolute term)

Most of the time you should be doing quadratic equations **to solve**. You're looking for numbers for the variable that makes up the equation **fulfill**. These numbers are called **solutions**. All solutions make up the **Solution set $$ L $$**.

In a quadratic equation, the variable x occurs in the 2nd power, but not in a higher power.

- It's about equations with one variable (usually x).
- to the power of 2 means "square".

**"Fulfill"**means: You insert a number for the variable in the equation and a true statement such as 2 = 2 emerges.- The
**solutions**quadratic equations are often infinite, non-periodic decimal fractions (irrational numbers).

### Simple quadratic equations

The simplest quadratic equations take the form

$$ x ^ 2 = r, r in RR $$.

The $$ r $$ is any real number.

#### Example:

$$ x ^ 2 = 9 $$ with $$ r = 9 $$

You can do other quadratic equations **equivalent transformations** bring it into this shape.

#### Example:

$$ 3x ^ 2 - 4 = 8 | + 4 $$

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

$$ x ^ 2 = 4 $$

The simplest quadratic equations contain terms with $$ x ^ 2 $$ and real numbers. They can be transformed into the form $$ x ^ 2 = r $$ $$ (rinRR) $$.

With an equivalent transformation, the solution set of the equation does not change!

### Solve simple quadratic equations

#### 1st example:

Solve the equation $$ x ^ 2 = 9 $$.

Solution:

$$ x_1 = 3 $$ and $$ x_2 = -3 $$, because $$ 3 ^ 2 = 9 $$ and $$ (- 3) ^ 2 = 9 $$.

Solution set: $$ L = {- 3; 3} $$

#### 2nd example:

Solve the equation $$ x ^ 2 = 1.69. $$

Solution:

$$ x_1 = 1.3 $$ and $$ x_2 = -1.3 $$,

because $$ 1.3 ^ 2 = 1.69 $$ and $$ (- 1.3) ^ 2 = 1.69. $$

Solution set: $$ L = {1.3; -1.3} $$

#### 3rd example:

Solve the equation $$ x ^ 2 = -4. $$

No solution, because $$ x ^ 2> 0 $$ for all real numbers x.

Solution set: $$ L = {} $$ (empty set)

If the quadratic equation is transformed into the form $$ x ^ 2 = r $$ and $$ r $$ is non-negative, the solutions of the equation can be determined by the square root of $$ r $$.

$$ x ^ 2 = 9 $$

$$ x_1 = + sqrt9 = 3 $$

$$ x_2 = - sqrt9 = - 3 $$

The square of a real number is always positive.

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### Form first

You can also solve more complicated equations if you can transform them into the form $$ x ^ 2 = r (r inRR) $$.

#### Example:

$$ 2x * (4-x) = 8 (x-1) $$

Forming:

Multiply the brackets on both sides.

$$ 2x * 4-2x * x = 8x-8 $$

$$ 8x-2x ^ 2 = 8x-8 $$ | $$ - 8x $$

$$ - 2x ^ 2 = -8 $$ | $$: (- 2) $$

$$ x ^ 2 = 4 $$ (purely square equation)

Solution:

$$ x_1 = 2 $$ and $$ x_2 = -2 $$

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

Sample:

$$ x_1 $$$$: $$ $$ 2 * 2 * (4-2) = 8 * (2-1) $$

$$4*2=8*1$$

$$8=8$$

Always try to simplify a given equation using equivalent transformation.

**Multiply out**: Each term in brackets is multiplied by the term in front of the brackets.

**sample**: Put the calculated solution in the variable.

### Solutions of the equation $$ x ^ 2 = r $$

What is the general solution?

An arbitrary equation of the form $$ x ^ 2 = r $$ is given.

Solutions: $$ x_1 = + sqrt (r) $$ and $$ x_2 = -sqrt (r) $$

The solvability of these equations only depends on the number $$ r $$.

There are 3 cases:

equation | number solutions | solution |
---|---|---|

$$ r> 0 $$$$: $$ $$ x ^ 2 = r $$ | 2 solutions | $$ x_1 = sqrt (r) $$ $$ x_2 = -sqrt (r) $$ |

$$ r = 0 $$$$: $$ $$ x ^ 2 = 0 $$ | 1 solution | $$ x = 0 $$ |

$$ r <0 $$$$: $$ $$ x ^ 2 = r $$ | no solution | $$———$$ |

$$ (sqrt (r)) ^ 2 = r $$ and $$ (- sqrt (r)) ^ 2 = r $$

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