Chapter 2: Quadratics

Question 1

4 ways of solving quadratic equations?

Answer 1

-Factorisation
-Solving without factorisation
-Completing the square
-Quadratic formula

Question 2

1st step in solving 2x+√x+1=0

Answer 2

Let √x=y

2y^2+y+1=0
Solve…

Question 3

For solving without factorisation method, solve:

(x-1)^2=5

Answer 3

x-1=+/-√5
x=+/-√5

Question 4

For ax^2+bx+c=0

How do I solve for x?

Answer 4

x=-b+/-√b2-4ac
2a

Question 5

When is the quadratic formula most appropriate to use?

Answer 5

-Co-efficient of x^2 is large
-3 parts are hard to be easily divided by a number

Question 6

Define completing the square.

Answer 6

Putting a quadratic equation in the form (x+a)^2+b=0/a(x+b)^2+c=0

Question 7

When is completing the square often used.

Answer 7

When x only appears once in the expression (ie (x+2)^2)

Question 8

Solve, using completing the square, 3x^2-18x+4=0

Answer 8

…
(x-3)^2-23/3=0
(x-3)^2=23/3
x-3=+/-√23/3
x=+/-√23/3 + 3

Question 9

Prove, using completing the square, that ax^2+bx+c=0

x=-b+/-√b2-4ac
2a

Answer 9

x^2/a+bx/a+c/a=0
(x+b/2a)^2-b^2/4a^2+c/a=0
(x+b/2a)^2=b^2/4a^2 - c/a
(x+b/2a)^2=b^2-4ac/4a^2
x+b/2a=+/-√b^2-4ac
2a
x=-b+/-√b2-4ac
2a

Question 10

Define the domain of a function.

Answer 10

Set of values of possible inputs of function.

Question 11

Define the range of a function.

Answer 11

Set of values of possible outputs of a function.

Question 12

Define the roots of a function.

Answer 12

Values of x when f(x)=0

Question 13

f(x)=2x-10
g(x)=x^2-9

a) Find g(5)
b) Find the values for x when f(x)=g(x)
c) Find the roots of f(x)
d) Find the roots of g(x)

Answer 13

a)
25-9=16
b)x^2-9=2x-10
x^2-2x+1=0
(x-1)(x-1)=0
x=1
c)2x-10=0
2x=10
x=5
d)x^2-9=0
(x+3)(x-3)=0
x=3 or x=-3

Question 14

How can you determine the value of the maximum/minimum of a function.

Answer 14

By completing the square

Question 15

(x-3)^2-7
What is the minimum value.

Answer 15

(3,-7)

Question 16

How do I know what the minimum value would be of a function, in terms of inputting values?

Answer 16

Putting in number that will produce lowest possible output, 0, and then output of this is the minimum value of function.

Question 17

How can you prove that completed-the-square versions of quadratics will always create minimum values, not maxima?

Answer 17

As anything squared=0

Therefore co-efficient of x^2 is positive.

Therefore u-shaped parabola.

Therefore turning point of quadratic will be a minimum point.

Question 18

Discuss the general shape of a quadratic graph.

Answer 18

-Depends on co-efficient of x^2
-If x^2 is greater than 0, then u-shaped parabola
-If x^2 is less than 0, then n-shaped parabola

Question 19

Y-intercept?

Answer 19

Value of y when x=0

Question 20

X-intercept?

Answer 20

Value of x when y=0

Question 21

Roots of a quadratic function?

Answer 21

Values of x when f(x)=0

Therefore as y=0, x-intercepts of graph.

Question 22

Minimum/maximum value of graph?

Answer 22

Highest/lowest point of curve, discovered by process of completing the square.

Question 23

What are the components of a sketch?

Answer 23

-General shape drawn, no specific points
-Axes values not written
-Only special co-ordinates of interest are usually included, like intercepts or POIs of multiple lines etc.

Question 24

Equation of line of symmetry is calculated how for a quadratic graph?

Answer 24

X-value of minimum=eqn. of line of symmetry

e.g.
Minimum (-3,-7)

Line of symmetry eqn. x=-3

Question 25

If the minimum point is (5,-2), what is the equation of the normal to this quadratic eqn.

Answer 25

Same as line of symmetry to curve,
minimum x-co-ordinate=5
Therefore, eqn. of normal to curve: x=5

Question 26

In the graph of y=ax^2+bx+c=0

The graph has a minimum at (7,-2) and passes thru. (8,0). Find a,b, and c.

Answer 26

Minimum (7,-2)
Therefore, line of symmetry x=7
For (8,0) reflection in line of symmetry x=7= (6,0)

Therefore, 0=(x-8)(x-6)
x^2-14x+48=0
a=1 b=-14 c=48

Question 27

For graphs with 2 roots that do not multiply to give y-intercept, how do we know what the eqn. is?

Exemplify.

Answer 27

Multiply by this as a scale factor
e.g. Roots=6,2
Y-intercept=3
3/12=1/4

Hence, 1/4(x-6)(x-2)
Expand.

Question 28

A line of ax^2+bx+c=0 passes through (6,0) with a minimum (5,-3). What are a,b,c.

HINT: At end of calc. when completed square, minimum occurs?

Answer 28

Minimum= (5,-3)
Line of symmetry: x=5

Reflection of (6,0) in line of symmetry x=5 is (4,0)

(x-6)(x-4)
=x^2-10x+24

Complete square:
(x-5)^2-25+24
(x-5)^2-1
Not give minimum (5,-3)

So, let scale factor=k
k(x-6)(x-4)

Sub (5,-3) into eqn:
-3=k(5-6)(5-4)
-3=k(-1x1)
-3=-k
k=3

Therefore, 3(x-6)(x-4)
3(x^2-10x+24)
=3x^2-30x+72

a=3,b=-30,c=72

Question 29

If the discriminant, b^2-4ac is greater than 0?

Answer 29

2 distinct, real roots.

Question 30

If discriminant=0

Answer 30

Equal, repeated roots

Question 31

Discriminant is less than 0?

Answer 31

No real roots.

Question 32

For a quadratic curve of 2 distinct, real roots?

Answer 32

Passes thru x-axis twice (when x=0 root occurs)

Question 33

For a curve of equal, repeated roots?

Answer 33

Touches it once, hence is tangent to x-axis.

Question 34

For a curve of no real roots?

Answer 34

Never touches/passes thru. the x-axis (minimum/maximum above/below the axis).

Question 35

Hence,

e.g. Find, using the discriminant, the values of K when L is a tangent to C.

Answer 35

If tangent, one solution, therefore repeated roots, b^2-4ac=0
…

Apply this to eqn of POI of L and C.

Question 36

Discuss the relation of p(x=q)^2+r=0 of r in this equation and the real roots of the equation.

Answer 36

If r is greater than 0, no real roots, with min point above the x-axis.
If r=0, there is equal, repeated roots, with the min point being on the x-axis (equation tangent to it).
If r is less than 0, two distinct real roots, min point below x axis, with equation passing thru. it twice.

Question 37

What is modelling in mathematics?

Answer 37

Utilisation of mathematical theory in real-life situations.

Question 38

Why are models used?

Answer 38

They can be used to solve a problem in a real-life situation, where normally some simplifying assumptions are applied.