Quadratic Part1

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Contents 1 Notes 1.1 Learning Objectives (Syllabus) 1.2 Prior Knowledge 2 Completing the Square 2.1 Preliminary 2.2 Guideline 2.3 Important 2.4 Examples 3 Exercise 1 4 To Find Max/Min 4.1 Value of x2 4.2 Max/Min value 4.3 To Find Max/Min value 4.4 Max/Min Point 5 Exercise 2 6 Sign of Quadratic Polynomial 6.1 To Show Sign of Quadratic Polynomial 6.1.1 Example 6.2 Maximum/Minimum Value & Sign 6.2.1 Example 7 Exercise 3

Notes

This is the one of the killer subtopics in the first half of the syllabus, second only to inequality. Just looking at it though, it doesn't seem like it, especially since we almost learn nothing new compared to add maths SPM. Two reasons :

• Anything that have already been covered in SPM, will have much more difficult questions in actual STPM, since it is pointless to issue SPM-level questions
• MOST students never actually did fully understand this topic in SPM. No, not even those with A1. Only a minority really understood the complexity of it. The rest just memorize formulas/workings.

Learning Objectives (Syllabus)

• use the process of completing the square for a quadratic polynomial
• derive the quadratic formula
• solve linear, quadratic, and cubic equations and equations that can be transformed into quadratic or cubic equations
• use the discriminant of a quadratic equation to determine the properties of its roots
• prove and use the relationship between the roots and coefficients of a quadratic equation

Prior Knowledge

• basic algebraic skills
• surds, complex numbers
• polynomials, factors, roots

Completing the Square

• This is one of the main skills that needs to be learnt well here. We will also need to master the various ways it can be used.
• I prefer to use a direct method of completing rather than using any formulas. It might seem difficult at first but you will get a hang of it.

Preliminary

Complete the following

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  • Since we can't write the on the left, we bring it over to the right and it becomes
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Guideline

Similarly

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Thus

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Important

• Rearrange and factorize (if needed) so that coefficient of is
• Final form is (need to multiply back if factored at the beginning)
• CHECK! The fastest way is check that it will give back the same constant term. This will check for most careless mistakes.

Examples

Complete the square

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• Already nicely arranged and the coefficient of is , so we can do directly
  • (Write out the form first, including the (so that we won't forget), and then only we fill in the blanks
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  • Check that gives us back

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• First, we will need to factor the
  • Be sure to factorize every term
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  • Once again, we will do it in one line, but fill in step by step
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  • Calculate the final constant (please use calculator!), remember to multiply back the
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  • CHECK!
    • . Use calculator to check it will give back
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• Needs to be factored too
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• Try it first! Answer is
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Exercise 1

Complete the square for the following

a)

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b)

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c)

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d)

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e)

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f)

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To Find Max/Min

Once we have completed the square, we can obtain the max/min value of the polynomial just by looking at the final form. This, like the direct method of completing the square, requires logical thinking rather than memorizing of steps.

Value of x²

For all real values of , we know that This is because

That is,

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Similarly

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Meanwhile

• Is it? Or ? It is
  • (This is because if CAN be zero, which is when , so it still will be )
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  • (Remember that both negative and positive values, when squared, will give us positive values, so even though is negative (when is positive), it does not matter)

That is,

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Note

Since for all real values of ,

• for all real values of
• Don't mix up with

Max/Min value

If , it means can take the value of , or any value greater than . Thus, we say that the minimum value of is

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To Find Max/Min value

Note : MUST complete the square first (if not already so)

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• Let's start with the simplest. There is no need to complete the square here.
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• This is slightly more complicated. Just keep in mind that we always start with and then we will work from there.
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  • (we can probably skip this step and straight to the max/min value step)
  • When does take this minimum value of ? When
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• Try this on your own first! (Writing the steps is as important as the value)
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• • (Can probably skip a step here)
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• Be careful with the negative here. Regardless, start with and work carefully from there
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• Try this on your own first!
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• Note that . DON'T just look at the , look at everything inside the bracket
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  • When is ?
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• Try this on your own!
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• Remember, don't memorize the steps, but understand the logic in each step, and you won't go wrong.
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  • When does take a minimum value of When
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Max/Min Point

If maximum/minimum value of when

• maximum/minimum point is

Exercise 2

For every question in Exercise 1 (let the each polynomial to be equals )

• a)State the maximum/minimum value of and the value of when it occurs
• b) the maximum/minimum point of its graph

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b)

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c)

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d)

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e)

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f)

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Sign of Quadratic Polynomial

Preliminary

• If ,
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• If ,
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• If could be positive or negative
• If could be positive or negative

Thus

• If has a positive minimum value
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  • or we say
• If has a negative maximum value
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  • or we say
• If it is neither, but we are asked to prove always positive/negative, it means we made a mistake in the completing the square

To Show Sign of Quadratic Polynomial

From above, we can see that we would need to find max/min value. Thus, we will first need to complete the square.

Important Note: Whenever question ask us to show something, never ever start with that something. That something must be the last line.

Example

• Show that for all
• First, remember, never start with what we need to show.
  • Thus, the working should look like this
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  • And NOT
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  • We do know that we need to complete the square of the LHS, so we will start from there
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    • Note again that it is WRONG to write
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  • So what we have till now is
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  • • Let's just start again from and see where we can go...
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    • Hmmm... but we don't need to prove that it is minimum , we need to prove that it is , so instead of this line, we will write
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    • Hurray... finished.... nope
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• Show that is always negative.
• Try this on your own first! Pay special attention to how you write the proof, as a wrong way will get you zero marks
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To find Max/Min vs To Prove Sign

Suppose we have , since

• If we need to find max/min,
• If we need to prove it is always positive,

Maximum/Minimum Value & Sign

• If given
• (it must have a positive minimum value)
• If given
• (it must have a negative maximum value)

Example

• Find range of if for all
• Be very aware that this is a given question, not a prove. Though it uses the same concept, the working is different (less rigid, actually)
  • We must use what is given to get a solution. Even if we can't see yet how it will give the answer, just write down what is given first
  • Let's complete the square
  • Note that we can complete the square directly from the inequality, it is not necessary to purposely write LHS=.... like we have to do if it was prove. Make sure you understand why.
  • We can see that the minimum value is , so this value must be positive
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Exercise 3

1) Prove that for all .

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2) Prove that for all .

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3) Prove that for all .

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4) Prove that is always positive for all .

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5) Find range of if for all .

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6) Given for all . Find range of .

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7) Show that if for all , .

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