Quadratics Flashcards
What is the shape of a quadratic graph?
A Parabola
A curve where any point is at an equal distance from a fixed point and a fixed line.
All quadratics have the same shape just squashed, streched or translated.
What is the quadratic form y = a(x-p)^2 + q
What does it tell us?
Completed square form
it tells us the line of symmetry is x = p and the max/min point is (p,q)
if its a max point, a will be -ve
if its a min point, a will be +ve
Quadratics in disguise
Some equations can be solved as quadratics
How would you solve:
x^4 - x^2 -6 = 0?
Let y=x^2, sub into original —>
y^2 - y - 6 = 0
(y -3)(y+2)=0, so y = 3 or y = -2
then change back —> x^2 = 3 or -2 (no real roots)
x = +/- root 3
How do you find roots of a quadratic or higher if it has roots?
Substitution
let y = x^2 then solve for y and reputting solution for y into y=x^2
Solving quadratics using the formula
1) complete the square –> (x-a)^2 = 0
2) Quadratic formula for ax^2 + bx + c = 0
1) (x-a)^2 = b –> x-a = +/- root b
2) -b +/- root b^2 - 4ac / 2a
The expression b^2 - 4ac is the discrimnant
what does this tell us?
The value of the discrimnant tells us how many times a quadratic graph intersects with the x axis.
when b^2 - 4ac < 0 graph doesnt cross x axis. Quadratic has no real roots
when b^2 - 4ac = 0 graph touches x axis once. Quadratic has equal roots
when b^2 - 4ac > 0 graph crosses x axis twice. Quadratic has two distinct roots.
What are the methods of solving linear simultaneous equations?
1) addition and subtraction
multiply one equation to get common coefficient of unknown in each then sutracts so there is only one 1 unknown left
2) substitution
rearrange to get one of the unknown in terms of the other then substitute into orignial equation
How do you find the point of intersection?
Solve them simultaneously
when must you reverse an inequaltiy sign during the solving of an inequality?
when you multiply or divide by a negative number, you must reverse the inequality sign.
you can check your inequality by inputting a value into the equation and seeing if it works.
How do you solve a quadratic inequality?
1) You factorise the quadratic expression which tells you the boundries of the solution
2) Then sketch the graph to detrmine the solution
If the quadraic is > 0, the solution CANNOT be written as a single inequality as x is not either and at the same time it is one or another.
What are the 3 different ways of solving a quadratic in the form ax^2+ bx + c
1) Factorise
2) Quadratic Formula
3) Completing the square
How do you find the roots of a quadratic equation?
put the quadratic equal to 0 and solve
How do you find the turning points of a graph (max/min)
Complete the square
How do you find the roots of a quadratic equation?
Put the equation equal to 0 and solve for x
