02.02 Completing the Square
In this lesson you will learn how to:
- Complete the square for quadratic expressions whose leading coefficient is \(1\)
- Complete the square for quadratic expressions whose leading coefficient is not \(1\)
- Solve quadratic equations by completing the square
- Find the maximum/minimum value of a quadratic function and the input value at which it occurs
Prerequisites from previous chapters:
- Expanding Brackets
- Factorising
- Surds
VIDEO LESSON
INTERACTIVE SELF-STUDY
Not all quadratic equations can be factorised nicely in order to solve them, however we can express all quadratics in a different form, called the completed square form.
Consider the quadratic expression \(x^2+bx\)
By factorising this as \(x(x+b)\), we can picture it as the area of a rectangle with lengths \(x\) and \(x+b\)
This rectangle can be split into a square of area \(x^2\) and rectangle of area \(bx\)
We can divide the rectangle of area \(bx\) into two identical rectangles of area \(\displaystyle\frac{b}{2}x\), by bisecting it through the side of length \(b\).
Now each smaller rectangle has lengths \(x\) and \(\displaystyle \frac{b}{2}\)
Then by shuffling the pieces around we can form a new bigger square with a small piece missing
i. What are the dimensions and hence the area of this new bigger square
ii. What are the dimensions and hence area of the small piece missing from the bigger squareiii. Hence form a relationship between \(x^2+bx\), the area of the new bigger square and the area of the missing piece
- We can see that \(\displaystyle x^2+bx≡\left(x+\frac{b}{2}\right)^2−\left(\frac{b}{2}\right)^2\)
In words, notice that you need to halve the coefficient of \(x\) term, add it to \(x\), square it, then subtract the square of that value - Hence \(\displaystyle x^2+bx+c≡\left(x+\frac{b}{2}\right)^2−\left(\frac{b}{2}\right)^2+c\)
- This is appropriately called the completed square form
Example 1 – (GRADE E)
Complete the square for the following expressions
a. \(x^2+10x\)
c. \(x^2−3x+7\).
NON-MONIC QUADRATIC EXPRESSIONS
Example 2 (Walkthrough) – (GRADE D)
Consider the quadratic expression \(2x^2+12x+5\)
Notice this time the coefficient of \(x^2\) is not equal to \(1\).
To get around this, we can just factor the \(2\) from the \(x^2\) and \(x\) term: \(2x^2+12x+5=2[x^2+6x]+5\)
We can then complete the square on \(x^2+6x\)
a. Complete the square for \(x^2+6x \)
b. Substitute your answer to part a back into the square brackets of \(2[x^2+6x]+5\) and hence complete the square for \(2x^2+12x+5\). (Make sure to expand properly!)- Whenever the coefficient of \(x^2\) is not equal to \(1\), simply factor it from the \(x^2\) and \(x\) term
e.g. \(\displaystyle ax^2+bx+c=a\left[x^2+\frac{b}{a}x\right]+c\)
Then complete the square on whatever is inside the square brackets
Example 3 – (GRADE D)
Write \(3x^2+12x+2\) in the form \(p(x+q)^2+r\), where \(p,q\) and \(r\) are integers to be found.
Example 4 – (GRADE D)
Complete the square for \(−x^2+6x−4\)
Example 5 – (GRADE C)
Complete the square for the expression \(ax^2+bx+c\)
- So we obtain that \(\displaystyle ax^2+bx+c≡a\left(x+\frac{b}{2a}\right)^2+\left(c-\frac{b^2}{4a}\right)\)
SOLVING QUADRATIC EQUATIONS
- From the completed square form, we can solve quadratic equations by rearranging it to make \(x\) the subject of the equation
Example 6 – (GRADE D)
Solve \(x^2+2x-4=0\). Give your answers in surd form
Example 7 – (GRADE C)
Solve \(16x^2+16x+1=0\). Give your answers in surd form
MINIMUM AND MAXIMUM VALUES OF QUADRATIC FUNCTIONS
We can also use the completed square form to find the maximum or minimum value of a quadratic function and the value of \(x\) at which it occurs
Suppose \(y=a(x−p)^2+q\), where \(a>0\)
i. What inequality can write down for \((x−p)^2\)
iii. Hence what inequality can you write down for \(a(x−p)^2+q\)
So we can see that for \(a>0\), \(y=a(x−p)^2+q≥q\).
In other words, for \(a>0\), \(y=a(x−p)^2+q\) has a minimum value of \(q\)
iv. At which value of \(x\) is this minimum value of \(y\) achieved?
- For \(a>0\), \(y=a(x−p)^2+q\) has a minimum value of \(q\), achieved at \(x=p\)
In words, the minimum value is the term after the square, and occurs at the value of \(x\) that makes the bracket equal \(0\)
Now suppose \(y=a(x−p)^2+q\), where \(a<0\)
i. What inequality can write down for \((x−p)^2\)
iii. Hence what inequality can you write down for \(a(x−p)^2+q\)
iv. Hence does \(y\) have a minimum or maximum value
v. At which value of \(x\) does \(y\) achieve its minimum/maximum value?
- For \(a<0\), \(y=a(x−p)^2+q\) has a maximum value of \(q\), achieved at \(x=p\)
In words, the maximum value is the term after the square, and occurs at the value of \(x\) that makes the bracket equal \(0\)
Example 8 – (GRADE C)
By completing the square, find the minimum value of \(2x^2−4x-3\) and the value of \(x\) at which this occurs. Explain how you know this is a minimum value rather than a maximum value.
CALCULATOR TIPS & TRICKS
The quadratic solver on your calculator can tell you the minimum/maximum value of a quadratic function. You can use this to ‘cheat’ your way to the completed square form in an exam.
Consider the quadratic \(3x^2-4x-5\)
When plugged into the calculator’s quadratic solver, it tells you the roots are \(\displaystyle \frac{2\pm\sqrt{19}}{3}\)
After this it tells you the value of \(x\) at which the minimum occurs is \(\displaystyle x=\frac{2}{3}\)
It then tells you that the minimum value of the function is \(\displaystyle y=-\frac{19}{3}\)
Putting this all together we have \(a=3\) (the coefficient of \(x^2\)), \(\displaystyle p=\frac{2}{3}\) and \(\displaystyle q=-\frac{19}{3}\)
Hence the completed square form is \(\displaystyle 3\left(x-\frac{2}{3}\right)^2 -\frac{19}{3}\)
Practice
Consider the quadratic expression \(5x^2+8x-1\)
i. Use your calculator to find the value of \(x\) at which the maximum occurs and the maximum value of the function
END OF LESSON!
You can now answer:
EDEXCEL Pure Year 1 Ex 2C – All Questions
EDEXCEL Pure Year 1 Ex 2D – All Questions