Nature of roots

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Nature of roots
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Investigate the Discriminant
In the formula, the sign of b2 4ac determines
the nature of the roots of the quadratic equation
We call b2 4ac the Discriminant
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Investigate the Discriminant
Using the excel applet, investigate what values
of a, b and c will lead to different types of
roots
Observe the number of x-intercepts, nature of
roots and their relationship with the value of
the discriminant
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Summary Worksheet Pg 4
The solutions given by
are directly determined by value of b2 4ac
When b2 4ac gt 0, the roots are real and
distinct
When b2 4ac 0, the roots are real and equal
When b2 4ac lt 0, the roots are not real/complex
Since b2 4ac determines the nature of roots of
the equation, we call it the discriminant
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Sketch
Case (1)
b2 4ac gt 0
a gt 0
a lt 0
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Sketch
b2 4ac 0
Case (2)
a gt 0
a lt 0
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Sketch
b2 4ac lt 0
Case (3)
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Question
What can you conclude if the roots of the
equation are real?
b2 4ac 0
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Observation
b2 4ac lt 0
Case (3)
y
a gt 0
a lt 0
y
ax2 bx c gt 0
x
x
ax2 bx c lt 0
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Observation
ax2 bx c gt 0 ? Case 3, no real roots
Entire curve lies above the x-axis
ax2 bx c lt 0 ? Case 3, no real roots
Entire curve lies below the x-axis
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Common Mistake
b2 4ac gt 0
ax2 bx c gt 0 ?
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Example TB Pg 60
x2 2kx (k 1)(k 3) 0 has no real roots.
Find the range of values of k.
Form an inequality with the correct condition
No real roots ? Condition
b2 4ac lt 0
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Example TB Pg 61
Show that the roots of x2 (1 p)x p 0 are
real and distinct.
Apply the correct condition
IF b2 4ac gt 0, then the roots are real and
distinct
Complete the square to show!
Since (p 1)2 is gt 0, therefore the roots are
real and distinct