Quadratics
I like to draw a smiley face (or crab claws) when multiplying out two pairs of brackets.
Some commentators said that the riots that swept parts of the country in 2011 were the result of mob rule. Well I have my own ‘mob rule’: times everything in one pair of brackets by everything in the other pair. And my mob stands for 'multiplying out brackets'. Take that, rioters!
If I multiply two brackets of the form (x + a)(x + b), I simply add a and b for the x-coefficient and multiply them for the number on the end.
Squaring an expression is the same as multiplying the expression by itself (as if it were two different expressions) and you were employing the smiley face method (or equivalent).
The FOIL technique is a favourite of mine for multiplying out two pairs of brackets: the letters stand for First, Outer, Inner, Last.
Quadratics get their quad prefix (met also in quadrilaterals and quad bikes where it means ‘four’) from the initial expansion having four terms before the inevitable simplification.
Two minuses make a plus (when you multiply them together).
Factorising into two pairs of brackets is the opposite of multiplying two pairs of brackets, just like factors and multiples were opposites at primary school.
When cancelling a fraction which has quadratic expressions for its numerator and denominator, the quadratic expressions will have one factor in common.
My eyes look like brackets so whenever I wear max factor eyeliner my boyfriend says I have max factor eyes (like factorise).
Check your factorization by multiplying out your answer.
If the final number of a quadratic expression is negative, then the two numbers you are looking for have a difference of the x-coefficient. And the larger of the two numbers will be the same sign as the x-coefficient.
If you’re stuck with a factorization, write out the factor pairs of the number at the end of the quadratic expression.
If every term has a common factor, then it’s one (pair of) bracket(s) factorising, but if there isn't a common factor it’s probably two (pairs of) brackets factorising.
If you have two terms which are squares (eg x-squared or 36) with a minus inbetween, then that’ll be the Difference of Two Squares.
Factorising applies to all letters of the alphabet, not just x.
Amaze your friends by checking their factorising – just multiply out their answers and compare with the original.
When factorising a quadratic, start by finding two numbers which multiply to make the number on its own, then see if it is possible to make the x-coefficient for each factor pair. Be careful with signs.
If you have a negative number on its own then you are looking for one positive and one negative number (and the sign of the larger of the two will be the same as the sign of the x-coefficient).
If the number on its own is positive, then the two numbers you seek are the same sign (i.e. both plus or both minus).
In Quantum Mechanics, the first bracket is called a Bra and the second a Ket. This is how I recall the word ‘bracket’.
When you have a number in front of the x-squared don’t be expecting to obey the ‘add up this and times to make this’ concept.
I always use two pairs of brackets when factorising a quadratic.
If I have a single x-squared, I find it useful to look for two numbers that add up to the x-coefficient and multiply to make the number on the end.
Before solving a quadratic equation, make sure it is equal to zero.
The area of a rectangle can be deduced by multiplying its base and height. In a similar way the perimeter can be deduced by doubling the sum of the base and height.
Quadratic equations must equal zero before you factorise.
If a quadratic expression is equal to zero then it can probably be factorised and the values of x which solve the equation determined.
If a bracket contains (ax – b), the answer is b over a.
When answering a quadratic equation that has arisen from a physical context, always refer back to the context once you have your solutions for x to determine whether either or both of the solutions are legitimate.
When children are old enough to forget how to factorise, they are told about the 'Quadratic Formula Song'. Treat this information delicately.
At the risk of making your brain explode, there is a third way of solving a quadratic equation, 'Completing the Square', lovingly crafted into a sweet tune originally about a cuddly bear made from fluff.
Some commentators said that the riots that swept parts of the country in 2011 were the result of mob rule. Well I have my own ‘mob rule’: times everything in one pair of brackets by everything in the other pair. And my mob stands for 'multiplying out brackets'. Take that, rioters!
If I multiply two brackets of the form (x + a)(x + b), I simply add a and b for the x-coefficient and multiply them for the number on the end.
Squaring an expression is the same as multiplying the expression by itself (as if it were two different expressions) and you were employing the smiley face method (or equivalent).
The FOIL technique is a favourite of mine for multiplying out two pairs of brackets: the letters stand for First, Outer, Inner, Last.
Quadratics get their quad prefix (met also in quadrilaterals and quad bikes where it means ‘four’) from the initial expansion having four terms before the inevitable simplification.
Two minuses make a plus (when you multiply them together).
Factorising into two pairs of brackets is the opposite of multiplying two pairs of brackets, just like factors and multiples were opposites at primary school.
When cancelling a fraction which has quadratic expressions for its numerator and denominator, the quadratic expressions will have one factor in common.
My eyes look like brackets so whenever I wear max factor eyeliner my boyfriend says I have max factor eyes (like factorise).
Check your factorization by multiplying out your answer.
If the final number of a quadratic expression is negative, then the two numbers you are looking for have a difference of the x-coefficient. And the larger of the two numbers will be the same sign as the x-coefficient.
If you’re stuck with a factorization, write out the factor pairs of the number at the end of the quadratic expression.
If every term has a common factor, then it’s one (pair of) bracket(s) factorising, but if there isn't a common factor it’s probably two (pairs of) brackets factorising.
If you have two terms which are squares (eg x-squared or 36) with a minus inbetween, then that’ll be the Difference of Two Squares.
Factorising applies to all letters of the alphabet, not just x.
Amaze your friends by checking their factorising – just multiply out their answers and compare with the original.
When factorising a quadratic, start by finding two numbers which multiply to make the number on its own, then see if it is possible to make the x-coefficient for each factor pair. Be careful with signs.
If you have a negative number on its own then you are looking for one positive and one negative number (and the sign of the larger of the two will be the same as the sign of the x-coefficient).
If the number on its own is positive, then the two numbers you seek are the same sign (i.e. both plus or both minus).
In Quantum Mechanics, the first bracket is called a Bra and the second a Ket. This is how I recall the word ‘bracket’.
When you have a number in front of the x-squared don’t be expecting to obey the ‘add up this and times to make this’ concept.
I always use two pairs of brackets when factorising a quadratic.
If I have a single x-squared, I find it useful to look for two numbers that add up to the x-coefficient and multiply to make the number on the end.
Before solving a quadratic equation, make sure it is equal to zero.
The area of a rectangle can be deduced by multiplying its base and height. In a similar way the perimeter can be deduced by doubling the sum of the base and height.
Quadratic equations must equal zero before you factorise.
If a quadratic expression is equal to zero then it can probably be factorised and the values of x which solve the equation determined.
If a bracket contains (ax – b), the answer is b over a.
When answering a quadratic equation that has arisen from a physical context, always refer back to the context once you have your solutions for x to determine whether either or both of the solutions are legitimate.
When children are old enough to forget how to factorise, they are told about the 'Quadratic Formula Song'. Treat this information delicately.
At the risk of making your brain explode, there is a third way of solving a quadratic equation, 'Completing the Square', lovingly crafted into a sweet tune originally about a cuddly bear made from fluff.
Straight Line Graphs
If I ever wish to find the midpoint of two points I simply find the average of the x-values and the average of the y-values.
The steeper a line, the larger the value of its gradient.
A vertical line has an infinite gradient and a horizontal line has a zero gradient.
The word ‘Parallel’ ends with a lower case l which is like a straight line [not in this font - mkyou2tube] (and the word itself contains two parallel l’s in the middle). The word ‘Perpendicular’, on the other hand, ends with a lower case r, which in a different font might look like it’s got a right-angle.
When I want the length of a line segment on a coordinate grid, I draw a right-angled triangle with the line segment as the hypotenuse and apply Pythagoras’ Theorem.
mkyou2tube et al used the theme of 'Y = MX + C' to raise bucketloads of lucre for the charity Children in Need.
Don’t write down values for the gradient (m) and y-intercept (c) until you have made y the subject.
M comes b4 C in da eqn of a str8 line as M comes b4 C in MC Hammer or McDonalds.
Straight line graphs are straight. Hence if you plot several points of what is equationalistically a straight line, but your plotted points fail to lie upon a single straight line, then you've done it wrong - don't blame the math.
The steeper a line, the larger the value of its gradient.
A vertical line has an infinite gradient and a horizontal line has a zero gradient.
The word ‘Parallel’ ends with a lower case l which is like a straight line [not in this font - mkyou2tube] (and the word itself contains two parallel l’s in the middle). The word ‘Perpendicular’, on the other hand, ends with a lower case r, which in a different font might look like it’s got a right-angle.
When I want the length of a line segment on a coordinate grid, I draw a right-angled triangle with the line segment as the hypotenuse and apply Pythagoras’ Theorem.
mkyou2tube et al used the theme of 'Y = MX + C' to raise bucketloads of lucre for the charity Children in Need.
Don’t write down values for the gradient (m) and y-intercept (c) until you have made y the subject.
M comes b4 C in da eqn of a str8 line as M comes b4 C in MC Hammer or McDonalds.
Straight line graphs are straight. Hence if you plot several points of what is equationalistically a straight line, but your plotted points fail to lie upon a single straight line, then you've done it wrong - don't blame the math.
Linear Equations
A linear equation (with no power greater than one) is so-called as to distinguish it from a quadratic equation (with powers of two).
If an equation has a fraction (or multiple fractions) and you wish to adapt the equation such that there is no longer a fraction, multiply every term by the denominator(s).
If both sides of a linear equation contain a single x-squared, you can cancel them out.
Contrary to popular belief, it is possible to have an answer to an equation which isn’t a positive integer. Examples include negative integers and fractions (note that it is forgivable and in fact popular to leave answers as top-heavy fractions (and not decimals)).
If ax = b, then x = b/a.
If an equation has a fraction (or multiple fractions) and you wish to adapt the equation such that there is no longer a fraction, multiply every term by the denominator(s).
If both sides of a linear equation contain a single x-squared, you can cancel them out.
Contrary to popular belief, it is possible to have an answer to an equation which isn’t a positive integer. Examples include negative integers and fractions (note that it is forgivable and in fact popular to leave answers as top-heavy fractions (and not decimals)).
If ax = b, then x = b/a.
Simultaneous Equations
A straight line can miss a circle or just gently touch it or pass through twice. If you are asked for the points of intersection there are two.
The two pairs of solutions for simultaneous equations (one linear, one quadratic) can be written as pairs of coordinates.
Always substitute the linear into the quadratic. You might need to rearrange the linear first.
Don’t forget to work out x once you've worked out y (or y once you've worked out x).
When squaring ax + b, do not forget to put the expression in brackets before multiplying it out.
Before embarking upon a factorisation it is always worth checking whether you can divide through by a common factor (e.g. 3).
For those people meeting simultaneous equations for thew very first time and with issues related to insomnia, the 'Same Sign Subtract' and 'Different Sign Add' could be a valuable educational cure.
The two pairs of solutions for simultaneous equations (one linear, one quadratic) can be written as pairs of coordinates.
Always substitute the linear into the quadratic. You might need to rearrange the linear first.
Don’t forget to work out x once you've worked out y (or y once you've worked out x).
When squaring ax + b, do not forget to put the expression in brackets before multiplying it out.
Before embarking upon a factorisation it is always worth checking whether you can divide through by a common factor (e.g. 3).
For those people meeting simultaneous equations for thew very first time and with issues related to insomnia, the 'Same Sign Subtract' and 'Different Sign Add' could be a valuable educational cure.
Graph Tranformations
No lion can sleep tonight when the 'Graph Transformations Song' is playing.