Chapter 2- Quadratics and cubics (Pure Maths)

Question 1

what’s the general formula of a quadratic?

Answer 1

the general formula of a quadratic is ax² + bx + c = 0

where a,b and c are constants and a is not 0

Question 2

what’s the quadratic formula? how to use it?

Answer 2

the quadratic formula is:

(-b ± √b² - 4ac ) / 2a

example:

2x²-4x-3 = 0
sub in to equation, a= 2, b = -4 and c= -3
then simplify :)

Question 3

how to complete the square? e.g x² + 6x + 3 = 0

Answer 3

x² + 6x + 3 = 0
(x² + 6x)- 9= 0
x+3 = ±√6
x= -3 ± √6

Question 4

what’s the discriminant?

Answer 4

the discriminant is b²- 4ac

Question 5

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

Answer 5

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

Question 6

how to find the Y intercept of a graph?

Answer 6

to find the Y intercept of a graph set x as zero

Question 7

how to find the X intercept of a graph?

Answer 7

to find the X intercept of a graph set Y as zero

Question 8

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 8

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 9

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

Answer 9

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

Question 10

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

Answer 10

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 11

how to factorise a cubic?

Answer 11

for a cubic X³+ bX²+ cX= X(X²+bX+c)

then solve as normal :)

Question 12

What’s the remainder theorem?

Answer 12

The remainder Theorem states:

When you divide f(x) by (x-a), the remainder is f(a)
when you divide f(x) by (ax-b), the remainder is f(a/b)

Question 13

what’s the factor theorem?

Answer 13

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

Question 14

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

Answer 14

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)

nx²-x²= 6x²
n=7

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