1. Maths revision course by Miriam Hanks
Quadratics – What are they?
x2 + 3x – 4
y2 – 4y + 2
1 – x2
Quadratics are equations where the highest power is 2.
What shape do their graphs make?
Maths revision course by Miriam Hanks

2. Maths revision course by Miriam Hanks
Quadratic graphs
Quadratics make a shape called a parabola
It is a smiley face if the coefficient of x2 is positive, or a sad face if it’s negative.
Maths revision course by Miriam Hanks

3. Maths revision course by Miriam Hanks
Quadratics – Sketching graphs
Any of these will help:
Identify shape: or
Find where it crosses the y-axis by putting x = 0
Find where it crosses the x-axis by putting y = 0
Complete the square to find the turning point
Differentiate to find the turning point
Maths revision course by Miriam Hanks

4. Maths revision course by Miriam Hanks
Completing the square
If there is a coefficient of x2, take it out as a factor of the first 2 terms.
eg 2x2 + 12x – 5
= 2[x2 + 6] – 5
Now insert a new bracket, move the “squared” sign to the outside of it, halve the number inside, and square and subtract it:
= 2[(x + 3)2 - 9] - 5
Maths revision course by Miriam Hanks

5. Maths revision course by Miriam Hanks
Completing the square
Next remove the outer bracket, remembering to multiply by your factor
= 2(x + 3) – 5
Finally, tidy up the last two terms:
= 2(x + 3)2 - 23
Maths revision course by Miriam Hanks

6. Maths revision course by Miriam Hanks
Completing the square
y =2(x + 3)2 - 23
How does this help draw the graph?
The turning point is
(-3, -23)
Maths revision course by Miriam Hanks

7. Maths revision course by Miriam Hanks
Solving a quadratic by using the formula.
The quadratic formula is not given to you in the Higher maths exam, so you should learn it:
Maths revision course by Miriam Hanks

8. Maths revision course by Miriam Hanks
The discriminant = b2 – 4ac
If b2 – 4ac = 0, then there is one real root (or two equal roots) Turning point is on x-axis
If b2 – 4ac > 0, then there are 2 real roots
Graph crosses x-axis twice
If b2 – 4ac < 0, then there are no real roots
Graph does not cross the x-axis
Maths revision course by Miriam Hanks

9. Quadratic inequalities
If your quadratic has a > or < sign,
Start solving it as normal, but when you get the 2 solutions, decide on the direction of the < or > arrows by drawing a diagram.
If you don’t draw the diagram, you will not get full marks in the exam.
Maths revision course by Miriam Hanks

10. Quadratic inequalities example 1
Eg Solve x2 + 3x – 4 < 0 gives
(x + 4) (x – 1) and so we mark x = -4 and
x = 1 on the diagram:
Since the original equation had a “< 0”, we look at where the graph is below the x-axis. Final answer:
-4 < x < 1 -4 1
Maths revision course by Miriam Hanks

11. Quadratic inequalities example 2
Eg Solve x2 + 3x – 4 > 0 gives
(x + 4) (x – 1) and so we mark x = -4 and x = 1 on the diagram:
Since the original equation had a “> 0”, we look at where the graph is above the x-axis. Final answer:
x < -4 and x > 1 -4 1
Maths revision course by Miriam Hanks

12. Maths revision course by Miriam Hanks
Quadratics in real life
Quadratics are used to make satellite dishes, suspension bridges and torches, all of which have a parabolic curve.
                                                                                                                                                                            
Maths revision course by Miriam Hanks

13. Maths revision course by Miriam Hanks
Quadratics in real life
When you throw an object, the path it takes is also a parabola:
Click for video clip
Maths revision course by Miriam Hanks