# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-functions-quadratic-models.html


## Quadratic Functions

"Not a lot of people know this, but I'm very good at mathematics. When I was an angry teenager, I used to sit in my room and do quadratic equations to calm myself down " Samantha Bond


#### Quadratic formula


##### What you have to know:

- You must be able to manipulate algebraic expressions. (If not, check this lesson)
- You must be able to expand and factorise the square difference and perfect square expressions. (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Another factorisation video
- 4. Check out this presentation

#### Special cases of quadratic formula


##### What you have to know:

- How to factorise a quadratic expression (If not, check this lesson)

##### Key Points

- 1. Watch the video:

#### Quadratic graph


##### What you have to know:

- How to plot linear graphs.

##### Key Points

- 1. Watch the video:

#### Vertex Form


##### What you have to know:

- How to expand and factorise perfect squares.

##### Key Points

- 1. Watch the video:

#### Extra


##### Material and references:

- Hodder Book SL(ISBN: 9781510462359) : 15A, 15B, 15C

##### Key Points

- 1. The quadratic graph \( y = a(x - p)(x - q) \) crosses the \( x \)-axis at \( (p,0) \) and \( (q,0) \). The quadratic graph \( y = a(x - h)^2 + k \) has vertex \( (h,k) \) and line of symmetry \( x = h \).
- 2. You should be able to complete the square: \[ x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c. \]
- 3. You should be able to solve quadratic equations by factorizing, completing the square, and using the quadratic formula. If \( ax^2 + bx + c = 0 \), then \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
- 4. You should be able to solve quadratic inequalities.
- 5. You should be able to use the discriminant \( \Delta \) to determine the number of roots of a quadratic equation, where \( \Delta = b^2 - 4ac \). If \( \Delta > 0 \), there are two distinct real roots; if \( \Delta = 0 \), there are two equal real roots; and if \( \Delta < 0 \), there are no real roots.

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