Chapter 2: Quadratics Flashcards
4 ways of solving quadratic equations?
-Factorisation
-Solving without factorisation
-Completing the square
-Quadratic formula
1st step in solving 2x+√x+1=0
Let √x=y
2y^2+y+1=0
Solve…
For solving without factorisation method, solve:
(x-1)^2=5
x-1=+/-√5
x=+/-√5
For ax^2+bx+c=0
How do I solve for x?
x=-b+/-√b2-4ac
2a
When is the quadratic formula most appropriate to use?
-Co-efficient of x^2 is large
-3 parts are hard to be easily divided by a number
Define completing the square.
Putting a quadratic equation in the form (x+a)^2+b=0/a(x+b)^2+c=0
When is completing the square often used.
When x only appears once in the expression (ie (x+2)^2)
Solve, using completing the square, 3x^2-18x+4=0
…
(x-3)^2-23/3=0
(x-3)^2=23/3
x-3=+/-√23/3
x=+/-√23/3 + 3
Prove, using completing the square, that ax^2+bx+c=0
x=-b+/-√b2-4ac
2a
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
Define the domain of a function.
Set of values of possible inputs of function.
Define the range of a function.
Set of values of possible outputs of a function.
Define the roots of a function.
Values of x when f(x)=0
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)
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
How can you determine the value of the maximum/minimum of a function.
By completing the square
(x-3)^2-7
What is the minimum value.
(3,-7)
How do I know what the minimum value would be of a function, in terms of inputting values?
Putting in number that will produce lowest possible output, 0, and then output of this is the minimum value of function.
How can you prove that completed-the-square versions of quadratics will always create minimum values, not maxima?
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.
Discuss the general shape of a quadratic graph.
-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
Y-intercept?
Value of y when x=0
X-intercept?
Value of x when y=0
Roots of a quadratic function?
Values of x when f(x)=0
Therefore as y=0, x-intercepts of graph.
Minimum/maximum value of graph?
Highest/lowest point of curve, discovered by process of completing the square.
What are the components of a sketch?
-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.
Equation of line of symmetry is calculated how for a quadratic graph?
X-value of minimum=eqn. of line of symmetry
e.g.
Minimum (-3,-7)
Line of symmetry eqn. x=-3
If the minimum point is (5,-2), what is the equation of the normal to this quadratic eqn.
Same as line of symmetry to curve,
minimum x-co-ordinate=5
Therefore, eqn. of normal to curve: x=5
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.
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
For graphs with 2 roots that do not multiply to give y-intercept, how do we know what the eqn. is?
Exemplify.
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.
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?
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
If the discriminant, b^2-4ac is greater than 0?
2 distinct, real roots.
If discriminant=0
Equal, repeated roots
Discriminant is less than 0?
No real roots.
For a quadratic curve of 2 distinct, real roots?
Passes thru x-axis twice (when x=0 root occurs)
For a curve of equal, repeated roots?
Touches it once, hence is tangent to x-axis.
For a curve of no real roots?
Never touches/passes thru. the x-axis (minimum/maximum above/below the axis).
Hence,
e.g. Find, using the discriminant, the values of K when L is a tangent to C.
If tangent, one solution, therefore repeated roots, b^2-4ac=0
…
Apply this to eqn of POI of L and C.
Discuss the relation of p(x=q)^2+r=0 of r in this equation and the real roots of the equation.
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.
What is modelling in mathematics?
Utilisation of mathematical theory in real-life situations.
Why are models used?
They can be used to solve a problem in a real-life situation, where normally some simplifying assumptions are applied.
