Mr. Jones, Heritage Academy
MPM2D – Principles of Mathematics 10
The Quadratic Formula
• So far we have covered finding the x-intercepts by factoring and by
graphing.
• Sometimes quadratic equations cannot be factored; in this situation we
can use the Quadratic Formula to find the x-intercepts.
• There are not always Real roots to a quadratic equation.
• Parabola A has 2 Real
Roots.
• Parabola B has 1 Real
Root.
• Parabola C has No Real
Roots.
The Quadratic Formula
−" ± √" % − 4'(
=
)*+,+', ", and(-./+0,./:2 = ' % + " + (
2'
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MPM2D–PrinciplesofMathematics10
Mr.Jones,HeritageAcademy
Example 1:
Solution
a) 2 % + 9 + 6 = 0
• From the equation we can see that ' = 2, " = 9 and ( = 6.
−" ± √" % − 4'(
=
2'
−9 ± 69% − 4728768
=
2728
=
−9 ± √81 − 48
4
=
−9 ± √33
4
∴ the Real Roots are =
<=>√??
@
and =
<=<√??
@
.
∴ the Roots can be approximated by ≅ −0.81 and ≅ −3.69.
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MPM2D–PrinciplesofMathematics10
Mr.Jones,HeritageAcademy
• Graphically, we can see that the Roots are the same as above.
Page 3 of 8
MPM2D – Principles of Mathematics 10
Mr. Jones, Heritage Academy
b) 4 % − 12 = −9
• This can be rewritten as:
4 % − 12 + 9 = 0
• From the equation we can see that ' = 4, " = −12 and ( = 9.
−" ± √" % − 4'(
=
2'
=
−7−128 ± 67−128% − 4748798
2748
=
12 ± √144 − 144
8
=
12 ± √0
8
=
12
3
BC
8
2
?
∴ the Real Root is = .
%
Page 4 of 8
MPM2D–PrinciplesofMathematics10
Mr.Jones,HeritageAcademy
• Graphically, we can see that the Root is the same as above.
Page 5 of 8
MPM2D – Principles of Mathematics 10
Mr. Jones, Heritage Academy
Example 2:
Solution
• From the equation we can see that ' = −5, " = 8 and ( = −3.
−" ± √" % − 4'(
=
2'
−8 ± 68% − 47−587−38
=
27−58
=
−8 ± √64 − 60
−10
=
−8 ± √4
−10
=
−8 ± 2
−10
• The x-intercepts are:
∴ D =
D =
D =
−8 + 2
−10
−6
−10
3
5
∴ % =
% =
−8 − 2
−10
−10
−10
% = 1
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MPM2D–PrinciplesofMathematics10
Mr.Jones,HeritageAcademy
• The x-value of the Vertex is found by adding the x-intercepts and then
dividing by 2.
FGHIGJ
3
+1
=5
2
FGHIGJ
3 5
+
5
5
=
2
FGHIGJ
8
=5
2
FGHIGJ =
FGHIGJ =
8 1
×
5 2
4
5
@
• The y-value of the Vertex is found by plugging in FGHIGJ = into the
L
quadratic equation.
2 = −5 % + 8 − 3
4 %
4
2 = −5 M N + 8 M N − 3
5
5
16
4
2 = −5 M N + 8 M N − 3
25
5
2 =
−16 32 15
+
−
5
5
5
2 =
1
5
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MPM2D–PrinciplesofMathematics10
@
D
Mr.Jones,HeritageAcademy
@
• Therefore the Vertex is O , P and the axis of symmetry is = .
L L
L
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