### task - an inquiry to solving all quadratic equations ^{1}:

$${ax^2 + bx + c = 0}$$

**Solve** each of the following equations.

As a general rule, each equation has got **two (!) solutions**.

**Write down** all your thoughts in your notes and **comment** about how the question differs from the previous one. Make sure not to expand the term in exercise **no 5** and use **no 5** to solve **no 6**.

**You do not have to solve all at once. You will be given plenty of time.**

Where you have square roots, make sure that

- there is
**no square root in the denominator**, i.e. rationalise the denominator. - you
**simplify each square root**as far as possible.

### questions

#### no 1

$${x^2 = 4}$$

#### no 2

$${x^2 - 3 = 0}$$

#### no 3

$${2x^2 - 1 = 0}$$

#### no 4

$${x^2 = 6}$$

#### no 5

$${(x + 2)^2 = 6}$$

#### no 6

$${x^2 - 6x + 9 = \frac{25}{4}}$$

#### no 7

$${x^2 -6x = 31}$$

#### no 8

$${x^2 + 4x = - \frac{7}{4}}$$

#### no 9

$${x^2 - \frac{2}{3}x = - \frac{1}{9}}$$

#### no 10

$${x^2 - 3x = - \frac{25}{4}}$$

#### no 11

$${2x^2 + 4x - 7 = 0}$$

#### no 12

$${\frac{1}{6}x^2 - \frac{1}{4}x - \frac{1}{6} = 0}$$

#### no 13

$${x^2 + 2px + q = 0}$$

#### no 14

$${ax^2 + bx + c = 0}$$

Solving question no 14 leads to the quadratic formula.

### different question no 13 for the pq-formula derived in Berlin

#### no 13b (berlin)

$${x^2 + px + q = 0}$$

*Towards New Teaching in Mathematics*, Issue 5, Peter Gallin, The Quadratic Equation – This is a Topic to Be Taught in an Inquiry-Based Way After All, Bayreuth, Germany, 2011, translation - Maren Distel, Singen^{[return]}