Quadratic Theory
Higher Maths
Quadratic Theory
The quadratic graph
Using the discriminant
Ans
Quadratic theory examples
Basic skills questions
Ans
Problem solving questions
Past paper questions
Click on a topic
Ans
The quadratic graph y = ax2+bx +c
In each of the diagrams below state whether
(i) a>0 or a<0
(ii) b2-4ac<0 or b2-4ac>0 or b2-4ac=0
Continued on next slide
Using the Discriminant
In the solution of the quadratic equation
ax2 + bx + c = 0
the solutions are given by
x
b
(b
2
 4 ac )
2a
The quantity b2 - 4ac is important .
It is called the discriminant.
It can be used to tell what kind of roots the equation will have.
The table on the next slide investigates this idea.
Continued on next slide
Consider the following table. You are required to complete the table
f(x) = ax2 + bx + c
1. f(x) = 2x2 + 3x - 7
Number of roots
2
a
b
c
D
2
3
-7
65
2. f(x) = x2 + 2x + 5
3. f(x) = x2 - 6x + 9
4. f(x) = 3x2 + x - 4
5. f(x) = x2 - 7x - 2
6. f(x) = 2x2 + x + 6
7. f(x) = x2 + 4x + 4
8. f(x) = -2x2 + 3x +1
9. f(x) = 5x2 + 3x - 2
10. f(x) = -3x2 - x + 2
See next slide before starting table
You could use the graphic calculator for this work if you wish.
Enter the function in Y1.
Press ZOOM and select 6: ZStandard
Draw each graph and determine the number of roots the quadratic has.
Enter the values of a , b and c and evaluate D.
Examine the table carefully and make
some conjectures about the connection between
the value of b2 - 4ac and the number of
roots of the corresponding quadratic equation.
Quadratic Theory Examples
[ y = ax2+bx +c ]
1. Choose one of either
a > 0 or a < 0
and one of
b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
corresponding to each of the six graphs below.
Continued on next slide
2. Use the discriminant b2 – 4ac to find the nature
of the roots of the equations below.
a) 2x2 – 7x + 1 = 0
b) 5x2 + 2x + 2 = 0
c) 9x2 – 24x + 16 = 0
d) x2 + x + 7 = 0
e) 6x2 – x – 1 = 0
f) 3x2 + 2x + 5 = 0
3. Examine the discriminant to see if the roots of the
following equations are real, equal or imaginary.
a)
2x2 – 5x – 1 = 0
b)
x2 + x + 7 = 0
c)
3x2 – 18x + 27 = 0
d)
2x2 + x + 1 = 0
4. Find k given that each of the following equations has equal roots.
a)
x2 – 8x + k = 0
b)
kx2 – 12x + 9 = 0
c)
x2 + kx + 16 = 0
5.
Find m if x2 + 2mx + 9 = 0 has equal roots.
Continued on next slide
6.
Find p if x2 + (p + 1)x + 9 = 0 has real distinct roots.
7.
Find p if (p + 1)x2 – 2(p + 3)x + 3p = 0 has equal roots.
8.
find c if x2 + (x + c)2 = 8 has equal roots.
9.
Show that the roots of k(x + 1)(x + 4) = x are not real if
1
9
10.
11.
<k<1
Find m if x2 + (mx – 5)2 = 9 has equal roots.
If
x  4 x  10
2
2x  5
= n form a quadratic equation in x and show
that, for real x, n  – 3 or n  2.
Solutions on next slide
Quadratics
Solutions
1.
i) a < 0
iii) a > 0
v) a > 0
2.
a)
b)
c)
d)
e)
f)
b2 – 4ac > 0 ii)
b2 – 4ac = 0
b2 – 4ac 0
a > 0 b2 – 4ac < 0
iv) a < 0 b2 – 4ac = 0
vi) a < 0 b2 – 4ac < 0
D = 41 roots are real and distinct
D = – 36,
no roots
D = 0,
roots are equal
D = – 27,
no roots
D = 25 ,
roots are real and distinct
D = – 56, no roots
Continued on next slide
3.
a)
b)
c)
d)
(–5)2 – 4.2.(–1) = 33
12 – 4.1.7 = – 27
(–18)2 – 4.3.27 = 0
12 – 3.2.1 = – 7
4.
a)
b)
c)
(–8)2 – 4.1.k = 0 , 64 – 4k = 0 , k = 16
(–12)2 – 4.k.9 = 0 , 144 – 36k = 0 , k = 4
k2 – 4.1.16 = 0 , k2 = 64 , k =  8
5.
(2m)2 – 4.1.9 = 0 , 4m2 = 36 , m2 = 9 , m =  3
6.
(p + 1)2 – 4.1.9 > 0 , (p + 1)2 – 62 > 0 , (p – 5)(p + 7) > 0
p < – 7 or p > 5
7.
real roots
imaginary roots
equal roots
imaginary roots
[–2(p + 3)]2 – 4(p + 1)(3p) = 0
4(p2 + 6p + 9) – 12p2 – 12p = 0
4p2 + 24p + 36 – 12p2 – 12p = 0
2p2 – 3p – 9 = 0
(p – 3)(2p + 3) = 0
p = – 3/2 or p = 3
Continued on next slide
x2 + (x + c)2 = 8
2x2 + 2cx + c2 – 8 = 0
This has equal roots when (2c)2 – 4(2)(c2 – 8) = 0
4c2 –8c2 + 64 = 0
4c2 = 64
c=4
9.
k(x + 1)(x + 4) = x
kx2 + (5k – 1)x + 4k = 0
This has imaginary roots if (5k – 1)2 – 4(k)(4k) < 0
25k2 – 10k + 1 < 0
9k2 – 10k + 1 < 0
(k – 1)(9k – 1) < 0
1
<k<1
8.
9
Continued on next slide
10.
x2 + (mx – 5)2 = 9
(m2 + 1) – 10mx + 16 = 0
This has equal roots when (–10m)2 – 4(m2 + 1)(16) = 0
100m2 – 64m2 – 64 = 0
36m2 – 64 = 0
9m2 = 16
m =  4/3
11.
x2 + 4x + 10 = n(2x + 5)
x2 + (4 –2n)x + 10 – 5n = 0
This has real roots when (4 – 2n)2 – 4(1)(10 – 5n)  0
16 – 16n + 4n2 – 40 + 20n  o
4n2 + 4n – 24  o
n2 + n – 6  0
(n + 3)(n – 2)  0
n  – 3 or n  2
Quadratic theory - Basic skills questions
Basic skills - Solutions
Quadratic Theory – Problem solving questions
Quadratic Theory
Exam Level Questions/ Past Paper questions.
1. For what values of ‘p’ does the equation x2 – 2x + p = 0 have equal roots.
2. Show that the roots of the quadratic (k-2)x2 – (3k-2)x + 2k = 0
are always real.
3. If ‘k’ is a real number show that the roots of the equation
kx2 + 3x + 3 = k
are always real.
4. The roots of the equation (x+1)(x+k) = -4 are equal. Find the value of ‘k’.
5. Find the values of ‘k’ for which the
equation 2x2 + 4x – k = 0 has equal roots.
6. Calculate the least positive integer ‘k’
so that the graph shown does not cut
or touch the x axis.
(0,k)
y = kx2 -8x + k
7.
Show that the equation
(1-2k)x2 – 5kx - 2k = 0
has real roots for all integer values of ‘k’.
8. For what values of ‘k’ has the equation x2 – 5x + (k+6) = 0
have equal roots?
9. If f(x) = 2x+1 and g(x) = x2 + k, show that the equation
g(f(x)) – f(g(x)) = 0 reduces to 2x2 + 4x – k = 0 and find the
value of ‘k’ for which this equation has equal roots.
What kind of roots does this equation have when k=6?
10. For what values of ‘k’ does the equation 5x2 – 2x + k = 0 have real roots?
11. For what value of ‘a’ does the equation ax2 + 20x + 40 = 0
have equal roots?
12. Find ‘p’ given that the equation x2 + (px – 5)2 = 9, has equal roots.
13. Given that
x
2
 4 x  10
2x  5
and hence show that if n
of the equation are real.
n
 3
, form a quadratic equation in x
or n  2
then the roots
14. Find ‘m’ if the equation (2m-1)x2 + (m+1)x + 1 = 0 has equal roots.
If m lies between these values find the nature of the roots.
15. Show that the roots of the equation k(x+1)(x+4) = x are not real if
1
9
<k<1.
16. Find ‘k’ given that the equation kx2 + (2k+1)x + k = 0 has equal roots.
17. If ‘k’ is a real number, show that the roots of the equation
x
2
 2 x  21
3x  7
 2k
, are always real.
18. For what values of ‘k’ does the equation
x(x-4) + 2 = k(2x – 3k) have real roots?
19. Show that the line y = x + c meets the
parabola y = x2 – 3x where x2 – 4x – c = 0.
Find the value of ‘c’ if the line is a tangent to the parabola.
20. Find the value of ‘n’ if the equation
, is to have equal roots.
(x  2)
x
2
2
2
n
Answers – Exam level questions
Q1. p = 1
Q2. (k+2)2 is always greater than or equal to 0
because it is a quantity squared.
Q3. (2k-3)2 is always greater than or equal to 0
because it is a quantity squared.
Q4. k = 5, k = -3
Q5. k = -2
Q6. No roots if k<-4 or k>4 therefore the smallest
positive integer k is k = 5
Q7. b2 – 4ac = 0 when k=0 or k = -8/9 therefore real for
all integer values of k.
Q8. k = 0.25
Q9. k = -2 ; If k = 6 there are two real distinct roots.
Q10. k  1
5
Q11. a = 2.5
Q12. p = 4/3 , p = -4/3
Q13. n  2 or n  -3
Q14. m = 1, 5
1
Q15. No roots for
<k<1
9
Q16.
Q17.
Q18.
Q19.
Q20.
k = -1/4
Roots are not always real. The roots are real if k  2 or k
Real roots if k is between (1-2) and (1+2)
c = -4
n = 0 and n = 3
 1
1
9