Quadratics

Question 1

Factorising Quadratics,

common factor

grouping

difference of 2 squares

perfect squares

completing the square meaning

completing the square process

quadratic formula

Answer 1

If every term in the expression has a common factor, factor out the number first

Like terms may need to be grouped together to find a common factor

a^2 - b^2 = (a + b) (a - b)

a^2 - 2ab + b^2 = (a - b) (a - b) = (a - b)^2
a^2 + 2ab + b^2 = (a + b) (a + b) = (a + b)^2

a quadratic equation expressed in the form of a perfect square plus or minus a constant

1. half the constant of x^2 and place in the brackets
2. half the coefficient of x in the brackets
3. minus half the coefficient of x outside of the brackets
4. simplify the outside of the brackets

(-b ± √b^2 - 4ac) / 2a

Question 2

Testing for number of solutions,

discriminant meaning

discriminant formula

discriminant is greater than 0

discriminant equal to 0

discriminant less than 0

Answer 2

For quadratic equations when y = 0 the parabola cuts the x-axis at either 0, 1 or 2 points, the discriminant controls the number and nature of the solutions.

∆ = b^2 - 4ac

Two distinct real roots

One repeated real root

No real root

Question 3

Sum and product of roots,

sum rule

product rule

Answer 3

If ax^2 + bx + c = 0 has the roots α and β then:

α + β = -b / a

αβ = c / a

Question 4

Simultaneous equations,

sketching

elimination by addition or subtraction

elimination by substitution

Answer 4

Sketch the graph of each equation to find points of intersection, x intercept and y intercept

Done by adding or subtracting the equations to eliminate an unknown and may be done by multiplying an equation by a constant

Rearrange either equation to isolate a variable which the equation can then be substituted into the other for that to have only one variable and then be solved

Question 5

Equations that reduce to quadratic equations,

process for x^4

Answer 5

For a quartic function:

1. Place a new variable as x^2
2. Solve the quadratic fully with the new variable
3. Place x^2 as each solution
4. Solve by square rooting each solution with ±

Question 6

Sketching quadratic function parabolas,

factorisation process

completing the square form

Answer 6

Sketching parabolas with factorisation:

1. Factorise the function
2. x intercept found at y = 0
3. y intercept found at x = 0
4. Find halfway between the x intercepts, X3
5. Sub X3 into original equation to find corresponding y
6. X3 and its y value make the coordinates of the vertex
7. Sketch graph with title equation and each point found

Sketching parabolas with completed square form:
1. Express equation in completed square form
a(x + h)^2 + k
2. x intercept found at y = 0
3. y intercept found at x = 0
4. X3 (axis of symmetry) found at -h , X3 = = -h
5. Sub X3 into completed square form equation to find y
6. X3 and its y value make the coordinates vertex
7. Sketch graph with title equation and each point found

Question 7

Quadratic inequalities,

forms equation may be in

process when solving for inequality based on y

Answer 7

y = ax^2 + bx + c
y = a(x - p)(x - q)
y = a(x - r)^2 + s

1. Factorise the function
2. Find x intercept/s
3. Sketch graph (positive up or negative down)
4. Find values of x that satisfy the y inequality