Quadratic Equations And Cubic Equations Worksheet With Answers
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Quadratic Equations
1.1
Roots of a quadratic equation
The general form of a quadratic equation is
2
ax
+ bx + c = 0
(1.1)
where a, b, c are constants and a = 0.
The roots of the quadratic equation (1.1) is given by
b
b
2
4ac
b
D
x =
=
2a
2a
2
where D = b
4ac.
The nature of the roots of a quadratic equation is determined by the value
of D.
(i) If D > 0, the equation will have two different real roots.
(ii) If D = 0, the equation has two equal roots.
(iii) If D < 0, the equation has complex roots.
1.2
Sum and product of the roots
Let α and β be the roots of the quadratic equation (1.1), then it is equivalent
to the equation
(x
α)(x
β) = 0
or
2
x
(α + β)x + αβ = 0
(1.2)
Dividing (1.1) through by a, we have
b
c
2
x
+
x +
= 0
(1.3)
a
a
Comparing (1.2) and (1.3), we obtain
b
c
α + β =
,
αβ =
(1.4)
a
a
Using (1.4), we can find the sum and the product of the roots directly from
the coefficients in the quadratic equation (1.1).
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