Grade 11 Mathematics/Investigation
1
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Limpopo DoE/Term 1 2025
NATIONAL
SENIOR CERTIFICATE
GRADE 11
MATHEMATICS
INVESTIGATION
MARKS
: 100
This investigation consists of 10 Pages.
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Grade 11 Mathematics/Investigation
2
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INVESTIGATION: DERIVATION OF THE QUADRATIC FORMULA AND ITS
SIGNIFICANCE
The aim of this investigation is to establish the quadratic formula, allowing us to find
solutions to any quadratic equation of the form ax 2 + bx + c = 0 . Let’s start by using the
process of completing the square to solve the equation x 2 + 2 x − 8 = 0 . We could solve this
equation through factorisation but let’s use completing the square and colour code the terms
to follow the process. We can then try to follow the same process to solve ax 2 + bx + c = 0 .
This investigation is set out as a fill in the gaps investigation. It is advisable that this paper is
used as an answer sheet as well.
SECTION A: UNDERSTANDING COMPLETING THE SQUARE
[15]
1. Starting with x 2 + 2 x − 8 = 0 , move the constant term to the right-hand side:
We can represent this problem visually as the sum of the area of a square with side
length x and the area rectangle with side lengths 2 and x, is equal to a square with total
area 8:
1.1 Fill in the missing side lengths.
Figure 1: Blue
(3)
Figure 2: Green
Figure 3: Orange
Equation 1 illustration 1
Splitting the green rectangle down the middle we can place it on either side of the blue
rectangle.
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1.2 Fill in the missing side lengths and areas.
(5)
Figure 2: Orange
Figure 1: Blue & green
Equation 1 illustration 2: We can “complete the square” on the left by adding the missing corner.
We do this to both sides to balance the equation.
1.3 Fill in the missing area’s.
(2)
Figure 3: Yellow
Figure 1: Blue & green
Figure 2: Orange
Equation 1 illustration 3: Combining the terms on the right-hand side to form a new square, we
now have equivalent squares.
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1.4 Fill in the missing area for the new rectangle on the right.
(2)
Figure 2: Grey
Figure 1: Blue, green and yellow
Equation 1 illustration 4
And hence,
( x + 1) = 9
2
1.5 So, we know the side length x + 1 must be equal to the side length of the grey rectangle
which is 3, so a solution is x =
.
(2)
1.6 If we are looking to solve the equation generally and not just in a concrete case we can
also have x + 1 = −3 , and hence our second solution is x =
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(1)
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Grade 11 Mathematics/Investigation
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SECTION B: DEDUCING THE QUADRATIC FORMULA
[32]
Let’s now repeat this process to deduce the quadratic formula.
2. Starting with ax 2 + bx + c = 0 , divide through by a and move the constant term to the
right-hand side:
Let’s again represent this problem visually as the sum of the area of a square with side
length x and the area of a rectangle with side lengths
b
and x, is equal to a square with total
a
area −ac:
2.1 Fill in the missing side lengths and area.
Figure 1: Blue
(4)
Figure 2: Green
Figure 3: Orange
Equation 2 Illustration 1
Splitting the green rectangle down the middle we can place it on either side of the blue
rectangle.
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2.2 Fill in the missing side lengths and areas.
(6)
Figure 2: Orange
Figure 1: Blue & Green
Equation 2 Illustration 2
We can “complete the square” on the left by adding the missing corner. We do this to both
sides to balance the equation.
2.3 Fill in the missing areas.
(3)
Figure 3: Yellow
Figure 1: Blue, Green & Yellow
Figure 2: Orange
Equation 2 Illustration 3: Combining the terms on the right-hand side to form a new square, we
now have equivalent squares.
2.4 Fill in the gaps to find a simplified expression for the area of our new grey rectangle: (2)
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c b2
− + 2
a 4a
=−
Limpopo DoE/Term 1 2025
4ac b 2 (make both fractions have the same denominator)
+
4a 2 4a 2
b 2 − 4ac
=
4a 2
(combine to form a single fraction)
2.5 Does the numerator look familiar? If yes, what is it called?
(2)
2.6 Fill in the gap for the area of our new grey rectangle.
(1)
Figure 2: Grey
Figure 1: Blue, Green & Yellow
Equation 2 Illustration 4
b  b 2 − 4ac

And hence,  x +  =
2a 
4a 2

(2)
From here let’s rearrange algebraically:
(3)
2
a) Take the square roots of both sides, don’t forget to indicate there are two answers
________________
b) Simplify the fraction by recalling
x
=
y
x
y
_______________________
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c) Isolate the x on the left by moving the other term to the right
___________________________
d) Write the right-hand side as a single fraction
_______________________
If you have followed the steps correctly you have established quadratic formula, i.e. for a
quadratic equation of the form ax 2 + bx + c = 0 , solutions are:
2.7 x =
−b  b 2 − 4ac
2a
(2)
Further questions to ponder:
3. Who was the first person to establish this formula and when?
(2)
4. Are there other ways to establish the formula?
(5)
SECTION C: SIGNIFICANCE OF THE QUADRATIC FORMULA
[3]
The quadratic formula defines the points ( x;0 ) on the parabolic graph, where the parabola
y = ax2 + bx + c crosses the x -axis and it can be separated into two terms,
x=
−b  b 2 − 4ac
2a
x=−
b
b 2 − 4ac

2a
2a
The first term −
b
describes the (i)
2a
b 2 − 4ac
, gives the (ii)
2a
, the line x = −
b
. The second term
2a
the roots are away from the axis of symmetry.
If the parabola’s vertex is on the x -axis, then the corresponding equation has a single
repeated root on the line of symmetry, and this distance term is zero, algebraically,
the (iii)
commonly known as b 2 − 4ac = 0 .
(3)
GRAND TOTAL: 100 MARKS
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Grade 11 Mathematics/Investigation
Level 1 (1x)
Criteria
Section A: Understanding Completing the Square [36 Marks]
Missing side lengths
(1.1 & 1.2)
No attempt to fill inside
lengths. (0 correct)
Missing areas (1.2-1.6)
Incorrect areas with
significant errors. (0 correct)
9
Limpopo
DoE/Term 1 2025
NSC
9
No understanding of
balancing the equation and
adding missing corners.
(0 correct)
Section B: Deducing the Quadratic Formula [52 Marks]
Completing the square
(1.3-1.4)
Level 2 (2x)
Level 3 (3x)
Most side lengths completed
correctly.
(1-2 correct)
Some areas calculated correctly
but with noticeable errors.
(1-3 correct)
All side lengths accurately identified and
completed with clear understanding
(3-5 correct).
Partial completion of the square
with some understanding of
balancing both sides. (1 correct)
All correct completion of square and
correct addition of missing corners. (2-3
correct)
Most areas are correctly calculated and
logical reasoning shown. (4-5 correct)
LDoE/Term 1 2025
Level 4 (4x)
Max Marks
/x4
All areas are calculated correctly with
detailed explanations of process. (6 and
both conclusion values correct).
/x3
/x4
Fill-in side lengths and
areas (2.1 & 2.2)
No side lengths identified.
(0 correct)
3 side lengths completed
correctly/with minor errors.
5 side lengths are completed correctly
with some gaps.
All side lengths are completed accurately
and systematically.
/x3
Completing the square
visually (2.3 & 2.4)
No attempt to visually
represent the square or add
missing corners. (0 correct)
One correct understanding of
completing the square visually but
lacks clarity or accuracy.
Most visual representations and
additions are correct with minor errors.
(2-3 correct)
Clear and accurate visual representation
with logical reasoning (4-5 correct).
/x3
Familiarity of the
numerator (2.5)
No attempt/response to the
familiarity of the numerator.
(0 correct)
Attempted to respond to familiarity
of the numerator with yes only.
(1 correct)
Conclusion/algebraic
rearrangements
(2.6 & 2.7)
No understanding of algebraic
rearrangements. (0 correct)
1-2 correct rearrangement
attempted but with errors.
Attempted to respond to familiarity of the
numerator with yes and named it.
(2 correct)
Most algebraic steps are correct with
some inaccuracies.
(3-4 correct)
Most algebraic steps are correct with
some inaccuracies.
(5-8 correct)
/x3
Historical and alternative
methods (3 & 4)
No attempt to answer
historical and alternative
method questions. (0 correct)
1-2 correct response with limited
explanation or accuracy.
3-4 correct responses correct with some
gaps in detail.
5-7 correct responses correct with some
gaps in detail.
/x1
one interpretation of terms with
noticeable gaps or errors.
(1 correct)
Two terms are interpreted correctly with
minor errors.
(2 correct)
Both terms are interpreted correctly as
well as the discriminant. (3 correct)
/x3
/x4
Section C: Significance of the Quadratic Formula [12 Marks]
Interpretation of terms
(i-iii)
Incorrect interpretation of
terms and their significance.
(0 correct)
Total
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