1. Investigating the relationships between the
roots and the coefficients of quadratic
equations.
T- 1-855-694-8886
Email- info@iTutor.com
By iTutor.com

3.  Solve each of the following quadratic equations
a) x2 + 7x + 12 = 0
b) x2 – 5x + 6 = 0
c) x2 + x – 20 = 0
d) 2x2 – 5x – 3 = 0
 Write down the sum of the roots and the product of the roots.
 Roots of polynomial equations are usually denoted by Greek
letters.
 For a quadratic equation we use alpha (α) & beta (β)

4.  ax2 + bx + c = 0
a(x - α)(x - β) = 0 a = 0
 This gives the identity
ax2 + bx + c = a(x - α)(x - β)
 Multiplying out
ax2 + bx + c = a(x2 – αx – βx + αβ)
= ax2 – aαx – aβx + aαβ
= ax2 – ax(α + β) + aαβ
 Equating coefficients
b = – a(α + β) c = aαβ
-b/a = α + β c/a = αβ

5.  Use the quadratic formula to prove the results from the
previous slide.
-b/a = α + β c/a = αβ
a
acbb
2
42
a
acbb
2
42
a
b
a
b
a
acbb
a
acbb
2
2
2
4
2
4 22
a
c
a
ac
a
acbb
a
acbb
a
acbb
22
2222
4
4
4
)4(
2
4
2
4

6.  Find a quadratic equation with roots 2 & -5
-b/a = α + β c/a = αβ
-b/a = 2 + -5 c/a = -5 2
-b/a = -3 c/a = -10
 Taking a = 1 gives b = 3 & c = -10
 So x2 + 3x – 10 = 0
 Note: There are infinitely many solutions to this problem.
 Taking a = 2 would lead to the equation 2x2 + 6x – 20 = 0
 Taking a = 1 gives us the easiest solution.
 If b and c are fractions you might like to pick an appropriate
value for a.

7.  The roots of the equation 3x2 – 10x – 8 = 0 are α & β
1 – Find the values of α + β and αβ.
α + β = -b/a = 10/3
αβ = c/a = -8/3
2 – Find the quadratic equation with roots
3α and 3β.
 The sum of the new roots is
3α + 3β = 3(α + β) = 3 10/3 = 10
 The product of the new roots is 9αβ = -24
 From this we get that 10 = -b/a & -24 = c/a
 Taking a = 1 gives b = -10 & c = -24
 So the equation is x2 – 10x – 24 = 0

8. 3 – Find the quadratic equation with roots α + 2 and β + 2
 The sum of the new roots is
α + β + 4 = 10/3 + 4 = 22/3
 The product of the new roots is
(α + 2)(β + 2) = αβ + 2α + 2β + 4
= αβ + 2(α + β) + 4
= -8/3 + 2(10/3) + 4
= 8
 So 22/3 = -b/a & 8 = c/a
 To get rid of the fraction let a = 3, so b = -22 & c = 24
 The equation is 3x2 – 22x + 24 = 0

9.  The roots of the equation x2 – 7x + 15 = 0 are α and β.
 Find the quadratic equation with roots α2 and β2
α + β = 7 & αβ = 15
(α + β)2 = 49 & α2β2 = 225
α2 + 2αβ + β2 = 49
α2 + 30 + β2 = 49
α2 + β2 = 19
 So the equation is x2 – 19x + 225 = 0

10. Call us for more
information:
www.iTutor.com
1-855-694-8886
Visit