Chapter 2- Quadratics and cubics (Pure Maths)

Question 1

what’s the quadratic formula?

Answer 1

the quadratic formula is:

(-b ± √b² - 4ac ) / 2a

Question 2

how to tell how many roots are there from the discriminant?

Answer 2

discriminant > 0 there’s 2 roots
discriminant = 0 there’s 1 root
discriminant < 0 there’s no real roots

Question 3

what does a quadratic x² graph look like:

if the coefficient of X² is positive - what does the graph look like?

if the coefficient of X² is negative - what does the graph look like?

Answer 3

if the coefficient of X² is positive - the graph is U-shaped

if the coefficient of X² is negative - the graph is n-shaped

Question 4

For a graph with the equation y= p(x+q)² + r . what’s the vertex?

Answer 4

The vertex of y= p(x+q)² + r is (-q, r)

Question 5

For a graph with the equation y= p(x+q)² + r .How to tell if the vertex is a minimum or maximum?

Answer 5

For y= p(x+q)² + r

if p is less than 0, the graph is u shaped, so the vertex is a minimum

if p more than 0, the graph is n shaped, so the vertex is a maximum

Question 6

what’s the factor theorem?

Answer 6

if f(x) is polynomial, and f(a) = 0, then (x-a) is a factor of f(x)

Question 7

how to factorise a cubic when x isn’t a factor?

Answer 7

1) use the factor theorem ( by trial and error) to find one of the factors of the cubic

2) use your factor to find a quadratic that gives the cubic when you multiply by that factor

3) factorise the quadratic you found

example

x³ +6x² +5x -12

coefficients add to zero so (x-1) is factor

work out the term that gives x³ term (x-1) (x² +12)

put nx in (x-1)(x² +nx+ 12)

looking at the x² coefficients

nx²-x²= 6x²
n=7

(X-1)(X+3)(X+4)