# What is 2 4 8 16 2x2

### 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).

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!

#### 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 \$\$