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Quadratics

expressions of the form ax2+bx2+c

Quadratic Equations

form ax2+bx2+c=0 where "a" does not equal to zero. May have two, one, or zero solutions.

Linear Equations

equations of the form ax+b=0 where "a" does not equal 0 and only have one solution.

x2=k

x= positive or negative square root of k (k>0)

x= 0 (k=0)

there are no real solutions (k<0)

x= 0 (k=0)

there are no real solutions (k<0)

Null Factor Law

when the product of two or more numbers is zero, then at least one of them must be zero. If ab=0 then a=0 or b=0.

Null Factor Law Steps

1. If necessary, rearrange the equation so one side is zero.

2. Fully factorise the other side (usually LHS)

3. Use the Null Factor Law

4. SOlve the resulting linear equations

5. Check at least one of your solutions

2. Fully factorise the other side (usually LHS)

3. Use the Null Factor Law

4. SOlve the resulting linear equations

5. Check at least one of your solutions

Ilegal Cancelling

We must never cancel a variable that is a common factor from both sides of an equation unless we know that the factor cannot be zero.

Quadratic Formula

If ax2+bx+c=0 where "a" does not equal to zero, then x = -b ± √(b² - 4ac)/2a

Quadratic Function

relationship between two variables which can be written in the form y=ax+bx+c where "x" and "y" are the variables and a, b, and c are constants, "a" does not equal to zero. (using function notation, can be written as f(x) = ax2+b+c

Conic Sections

curves which can be obtained by cutting a cone with a plane. Ancient Greek mathematicians were fascinated by conic sections

Parabola

name comes from the Greek word for "thrown" because when an object is thrown, its path makes a parabolic arc. Examples are parabolic mirrors used in car headlights, heaters, radar discs, etc.

Simplest Quadratic Function

y = x2

- curve is a parabola and opens upwards

- no neg. y values, curve doesnt go below the x-axis

- curve is symmetrical about y-axis

- curve has turing point or vertex at (0,0)

- curve is a parabola and opens upwards

- no neg. y values, curve doesnt go below the x-axis

- curve is symmetrical about y-axis

- curve has turing point or vertex at (0,0)

x-intercept

value of "x" where graph meets the x-axis and are found by letting "y" be 0 in the equation of the curve ( easy to find when quadratic is in factorised form)

y-intercept

value of y where the graph meets y-axis and are found by letting "x" be 0 in the equation of the curve (constant term in quadratic function)

factorising to find x-intercepts

any quadratic function, the x-intercepts can be found by solving the equation ax2+bx+c=0.

Graph of any quadratic function

- is a parabola

- has turning point or vertex

- is symmetrical about a line of symmetry

- has turning point or vertex

- is symmetrical about a line of symmetry

line of symmetry

equation of y=ax2+bx+c is x=-b/2a