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ROOTS OF QUADRATIC EQUATIONS
Website: math.nie.edu.sg/bwjyeo/it
The purpose of this worksheet is to investigate the relationship between the discriminant and the
nature of the roots of a quadratic equation.
Section A: Investigation
1.
The LiveMath template shows the quadratic equation ax 2  bx  c  0 and the curve
y  ax 2  bx  c . Change the values of a, b and c to get the equations in the table below.
2.
Record the discriminant D  b 2  4ac , the values of the roots and the types of roots. Sketch
the curve also, including the x-axis.
No.
Quadratic Eqn
b 2  4ac
Values of
Roots
1.
x2  2x  2  0
12
2.73,
0.732
2.
2 x 2  5x  3  0
3.
 x 2  4x  3  0
4.
x 2  2x 1  0
5.
4x 2  4x  1  0
6.
 x 2  4x  4  0
7.
x2 1  0
8.
2x 2  x  1  0
9.
 2x 2  x  2  0
© Joseph Yeo
1
Type of Roots
2 distinct real roots?
2 equal real roots?
2 non-real roots?
Sketch of Curve
Section B: Induction
3.
Observe the discriminant D  b 2  4ac and the type of roots in the table above. How do you
identify the type of roots just by looking at the discriminant?
4.
Explain why the discriminant affects the type of roots for a quadratic equation.
Hint: x 
5.
 b  b 2  4ac
2a
Observe the type of roots and the curve in the table above. What relationship do you observe
between the type of roots and how the curve cuts the x-axis?
Section C: Application
6.
Using what you have learnt in the previous section, complete the following table without
using your computer. First calculate the value of b 2  4ac . Then infer the type of roots and
how the curve cuts the x-axis by drawing a sketch without solving the equation.
No.
Quadratic Eqn
1.
2 x 2  3x  2  0
2.
9x 2  6x  1  0
3.
 2x 2  x  3  0
4.
 y2  4y  1  0
© Joseph Yeo
b 2  4ac
Type of Roots
2 distinct real roots?
2 equal real roots?
2 non-real roots?
2
Sketch of Curve
ROOTS OF QUADRATIC EQUATIONS
Section A: Investigation
2.
No.
Quadratic Eqn
b 2  4ac
Values of
Roots
1.
x2  2x  2  0
12
2.73, 0.732
2.
2 x 2  5x  3  0
49
3, 0.5
3.
 x 2  4x  3  0
4
3, 1
2 distinct real
roots
4.
x 2  2x 1  0
0
1, 1
2 equal real
roots
5.
4x 2  4x  1  0
0
0.5, 0.5
2 equal real
roots
6.
 x 2  4x  4  0
0
2, 2
2 equal real
roots
7.
x2 1  0
4
i, i
2 non-real roots
8.
2x 2  x  1  0
7
0.250.661i,
0.25+0.661i
2 non-real roots
9.
 2x 2  x  2  0
15
0.25+0.968i,
0.250.968i
2 non-real roots
Section B: Induction
3.
If D > 0, then 2 distinct real roots;
if D = 0, then 2 equal real roots;
if D < 0, then 2 non-real roots.
© Joseph Yeo
3
Type of Roots
2 distinct real
roots
2 distinct real
roots
Sketch of Curve
4.



5.
 b  b2  4ac
.
2a
If D  b 2  4ac  0 , then b 2  4ac is not defined. So no real roots.
The roots of a quadratic equation are given by x 
If D  b 2  4ac  0 , then  b 2  4ac  0 and so x has only one value. Therefore two
equal real roots.
If D  b 2  4ac  0 , then  b 2  4ac  0 will have two real values. Therefore two
distinct real roots.
If 2 distinct real roots, the curve will cut the x-axis at 2 distinct points;
if 2 equal real roots, the curve will touch the x-axis at 1 point;
if 2 non-real roots, the curve will not cut the x-axis.
Section C: Application
6.
No.
Quadratic Eqn
b 2  4ac
Type of
Roots
Sketch of
Curve
Values of Roots
(for Teacher’s Info)
1.
2 x 2  3x  2  0
25
2 distinct
real roots
2, 0.5
2.
9x 2  6x  1  0
0
2 equal real
roots
 13 ,  13
3.
 2x 2  x  3  0
23
2 non-real
roots
0.25+1.2i, 0.251.2i
4.
 y2  4y  1  0
20
2 distinct
real roots
0.236, 4.24
Section D: Conclusion
7.
Suggestions for lessons learnt:
 Relationship between the discriminant and the type of roots of a quadratic equation
 Relationship between the discriminant and the number of times the corresponding
quadratic curve cuts the x-axis
© Joseph Yeo
4
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