Quadratics

… Quadratic expressions always feature a squared term, for example:

x² – 4

or…

3x² + 2x – 8

or…

8x² + 4x

So the general form of a quadratic expression is ax² + bx + c

Factorising quadratic expressions using the “1AM” method, for example:

1 A M

x² – 3x – 10

… When you need to factorise a quadratic which has a coefficient of 1 on the x² at the front (not 3x² for example), then look for two numbers which MULTIPLY to give the end number, and ADD to give the middle number. So for the above example:

Multiply to give -10 (start here and list the possibilities)

Add to give -3 (then find the pair of numbers from your list which add to give -3)

The two numbers would be -5 and +2

(x – 5)(x + 2)

… It’s always a good idea to check your factorisation by multiplying out the brackets and seeing if it works.

… You can factorise quadratic expressions which have a coefficient of MORE than 1 on the x² at the front (for example 3x²) by ONLY looking for two numbers which MULTIPLY to give the end number (the ADDING rule is not possible with this type!). This may require a bit of trial and error to find the numbers which work :).

… Using the ‘split factor’ method to factorise more tricky quadratic expressions where the coefficient of the x² term is 2 or more.

For example, factorise 3x² + 10x – 8

Step 1. First multiply the x squared coefficient and the constant together:

3 × -8 = -24

Step 2. Then find two numbers which multiply to give -24 and add to give +10

So -2 × 12 = -24, and -2 + 12 = 10

Step 3. Now split the middle x term into two parts using your numbers:

3x² – 2x + 12x – 8

Step 4. Now factorise the first and second pairs (note that the brackets are always the same):

x(3x – 2) + 4(3x – 2)

Step 5. Finally put the numbers into their own brackets

(x + 4)(3x – 2)

… Rearrange quadratic expressions so that the x-squared term is first, then the ‘x’ term, and then the number.

… Remember, that if the middle ‘x’ term is negative, then the two numbers will be…?

… Sometimes you may need to take out a common factor first, and then factorise:

For example, 6x² – 3x – 30 can first be written as:

3(2x² – x – 10), and then factorise the bracketed part using the split factor method.

… How to factorise ‘Difference of Two Squares’ expressions, for example:

4 – x² would factorise to:

(2 + x)(2 – x)

or

16y² – 9x² would factorise to:

(4y + 3x)(4y – 3x)

or a sneaky version:

2x² – 50

factorise normally first:

2(x² – 25)

then:

2(x + 5)(x – 5)

… You can solve quadratic equations by thinking about the two factorised brackets as ‘multiplying potatoes’. For example if we factorise a quadratic equation into:

(x – 2)(x + 5) = 0

Then either x – 2 = 0, or x + 5 = 0

So x = 2 or x = -5

A ‘simple’ type of quadratic with only 2 terms might be:

12x² – 9x = 0

This factorises into:

3x(4x -3) = 0

And so we have one potato giving 3x = 0, so the first solution is x = 0

Or

The other potato: 4x – 3 = 0

4x = 3

x = 3/4, our second solution.

Solving quadratic equations using the quadratic formula

… If, for example, a question in the calculator paper says,

“…give you answer to 3 significant figures”, or

“find the exact solutions to…”, or

“give your answer in the form of a + b√3”

then this is a clue that you’ll need to use the quadratic formula.

… First write down the formula and a=… b=… c=… (Remember the signs)

Then substitute each value into the formula using brackets.

… It can be useful to simplify the quadratic before putting the values of a, b and c into the formula. For example, can you divide every term by a common factor?

… The calculator can simplify the final surd answers for you :).

… Take care with the minus signs! Remember that the calculator thinks that “-7²” actually means -(7)² = -49… So you should always use brackets when squaring negative numbers on the calculator: (-7)² = 49

… The quadratic formula also works for hard quadratic equations involving large numbers or where the x² coefficient is higher than 1.

… It is an excellent idea to check your two final answers by putting each one into the original quadratic equation using the ‘ANS’ key on your calculator.

… Solving quadratic equations using your calculator

MENU and select “xy=0” by pressing “=”

2 (polynomial)

2 (degree, which is the power)

Now enter the values of a, b and c from the quadratic equation by typing in the numbers and pressing “=” to enter each number.

Press “=” to find the first and second x value solutions

The calculator will also give you the minimum or maximum coordinates of the curve.

Plotting quadratic curves and solving equations graphically

… Sometimes you are asked to plot a quadratic equation on graph paper and then solve an equation.

… Think of a quadratic equation as a function machine, f(x). The x value (horizontal axis) is the input and the y result (vertical axis) is the output.

… Use a table of values to calculate the x and y coordinates to plot a graph. For example:

y = x² + 2x – 3

Your table would look something like:

x** ** -2 – 1 0 1 2**x² ** 4 1 0 1 4**2x** -4 -2 0 2 4**-3 ** -3 -3 -3 -3 -3

y -3 -4 -3 0 5

… Use the natural curvature of your wrist movement to help you when drawing the curved line of a graph (avoid ‘sketching’ lines… They should only be a single line on the page – don’t worry if it is slightly wobbly!).

… Solving quadratic equations graphically, for example, if you have plotted the curve y = x² – 2x + 1, solve the equation:

x² – 2x + 5 = x + 7 (i.e. find the x values which work).

First write the equation to solve under the equation of the graph you have plotted:

y = x² – 2x + 1

x + 7 = x² – 2x + 5

Rearrange the equation you are given to solve so that one side of it looks like the graph you have plotted. Here we need to subtract 4 from both sides:

y = x² – 2x + 1

x + 3 = x² – 2x + 1

Now we can solve the equation graphically by finding the x coordinate of the intercepts between your plotted curve and the line y = x + 3

… This is effectively solving simultaneous equations graphically, by plotting both lines on graph paper and finding the coordinates of the intersection points.

… “Find the solutions (or roots) to the equation…” means find the x values which make the equation work.

… Solving two simultaneous equations graphically, for example, if you have plotted the curve y = x² – 2x + 1, solve the equations

y = x² – 2x + 1

and

y = x + 1

To do this, first draw the line y = x + 1 onto the same graph as y = x² – 2x + 1 and find the coordinates of the points where the two lines intersect.

Remember to be as accurate as you can with your graph drawing!

Solving quadratic inequalities

For example, solve x² + 2x – 3 > 0

First sketch the curve of y = x² + 2x – 3. To do this you will need to find where the curve crossed the x-axis (y = 0 and so you’ll need to solve x² + 2x – 3 = 0). It’s also worth remembering that a ‘happy’ curve has a positive x squared term and a ‘sad’ curve has a negative x squared term.

Then highlight the parts of the curve for which x² + 2x – 3 > 0. This is where the curve’s y value is > 0.

Find the sets of values of x which work for these sections of the curve.

Note:

- If there is ONE set of values of x which satisfies the inequality (e.g. a single drawn line on a number line), then you could write (for example) “x > -2 AND x < 3”. This can also be written as “-2 < x < 3”
- If there are TWO distinct sets of values for x , then you could write (for example) “x < -2 OR x > 3”.

Note that this CANNOT be written as “-2 > x > 3” because this statement implies an ‘AND’ and so is impossible!

… Remember that a square root always gives you both a positive and a negative answer.

… What is the shape of the graph y = x² ?

… You can solve simultaneous quadratic inequalities by solving each inequality separately. Then compare the answers (using a number line often helps too). For example:

If one of the inequality solutions is -1 < x < 4 and the other inequality gives the solution x ≥ 2, then the ‘overlap’ of x values is:

2 ≤ x < 4

… Set notation can be used to indicate an inequality, for example:

-1 < x ≤ 3

can be written as:

x = {0, 1, 2, 3}

Using Completing the Square for quadratic functions:

… The giveaway clue for completing the square is when the question mentions “…can be written in the form (x + p)² + q “, which is the classic completing the square format.

Here’s a good question you could put onto a flashcard:

“Express x² – 8x + 3 in the form (x + p)² + q where p and q are integers to be found.”

… Competing the square enables us to find the coordinates of the minimum or maximum vertex of a curve (the lowest or highest point), sometimes called the ‘turning point’ of the function.

… It also helps us to solve quadratic equations to find the coordinates where the curve crosses the x axis (where y = 0).

However, unless directed by the question, I would recommend using the quadratic formula (rather than using completing the square) if you are asked to solve an equation like 2x² – 6x + 3 = 0. This is because completing the square can quickly get complicated if fractions are involved…

… You may be asked to compete the square for expressions where the ‘a’ value on the x squared term is greater than 1. For example:

2x² – 8x + 3

2 [ x² – 4x ] + 3

2 [ (x + 2)² – 4 ] + 3

2 (x + 2)² – 8 + 3

2 (x + 2)² – 5

… If the coefficient on the squared bracket is negative, then the curve will be ‘unhappy’ with a maximum vertex. For example:

-2(x + 2)² – 5