quadratics (y1)

quadratic equation

ax^2 + bx + c = 0

a, b and c are real constants and a != 0

quadratic roots

the solutions to the quadratic equation

discriminant

b^2 - 4ac

the value indicates how many roots the quadratic equation has:

if b^2 - 4ac > 0, there are two distinct roots

if b^2 - 4ac = 0, there is one repeated real root

if b^2 - 4ac < 0, there are no real roots

factorising

-the quadratic equation is formatted into (nx ± a)(mx ± b) = 0

-each bracket is treated as if it equals zero

-the values of x equal the values of the roots

completing the square

-the quadratic equation is formatted into k(x+a/2k)^2 ± b = 0

-this is formatted to k(x+(a/2k))^2 = ± b

-then (x+(a/2k))² = b/k

-then x+(a/2k) = ±√(b/k)

-this gives x = ±√(b/k) - (a/2k)

-gives two values of x which equal the values of the roots

additional uses of completing the square

-to find the turning point on a quadratic graph

-to prove and/or show results using the fact that a squared term is always >= 0 i.e. k(x±a)^2 ± b always gives a result >= b

quadratic formula

(-b ± (b^2 - 4ac))/2a

gives the values of the roots

functions

functions take inputs, apply mathematical operations to it and output the value

usually denoted as f(x), or g(x) if f(x) is already specified

roots of functions

the values of x which cause an output of 0

commonly determined by solving the function's equation as quadratic(it may need to be converted first)

quadratic graphs

graphs plotted from quadratic functions, the line takes the shape of a parabola

parabola shape and QE

if a is positive the parabola will be u shaped, if it is negative it will be n shaped

x-intercept and QE

the roots of the quadratic equation equal the co-ordinates at which the parabola intersects the x axis

the number of roots determines whether the parabola crosses the x axis, merely touches it or does not touch it

y-intercept and QE

the y intercept of the parabola is equal to c

turning point and QE

the turning point of the parabola can be found by completing the square to k(x±a)^2 ± b where the co-ordinates equal (-a, b)