Quadratic Equation 1 Mind Map Quadratic Equation Identify QE Example of QE and non-QE General Form Value of a, b and c Solving QE Root of QE Substitution Intersection of Graph 3 Methods Factorisation Completing the Square Formula Forming QE from Roots x 2 – SoRx + PoR = 0 SoR = -b/a PoR = c/a A few example of SoR and PoR Type of Root Quadratic Equation In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is 2 0 ax bx c + += where a ≠ 0. (For a = 0, the equation becomes a linear equation.) The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term. Example of Quadratic Equation 2 2 2 2 2 5 0 1 6 3 6 3 0 0 x x x x x − = − = + = = Difference Between Quadratic Eqaution and Quadratic Function www.studyguide.pk
Quadratic Equation In mathematics, a quadratic equation is a
polynomial equation of the second degree. The general form is
2 0ax bx c+ + =
where a 0. (For a = 0, the equation becomes a linear equation.)
The letters a, b, and c are called coefficients: the quadratic
coefficient a is the coefficient of x2, the linear coefficient b is
the coefficient of x, and c is the constant coefficient, also
called the free term or constant term. Example of Quadratic
Equation
2
2
2
2
2 5 01 6 36 3 0
0
xx
x xx
= =+ ==
Difference Between Quadratic Eqaution and Quadratic Function
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Quadratic Equation
2
Solving Quadratic Equation 3 Methods: z Factorisation z
Completing The Square z Quadratic Formula Factorisation
Example Solve x2 + 5x + 6 = 0. Answer x2 + 5x + 6 = (x + 2)(x +
3) Set this equal to zero: (x + 2)(x + 3) = 0 Solve each factor: x
+ 2 = 0 or x + 3 = 0 x = 2 or x = 3 The solution of x2 + 5x + 6 = 0
is x = 3, 2 Completing The Square
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Quadratic Equation
3
Quadratic Formula
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Quadratic Equation
4
Forming Quadratic Equation from Its Roots
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Quadratic Equation
5
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Quadratic Equation
6
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Quadratic Equation
7
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Quadratic Equation
8
Nature of Roots of a Quadratic Equation The expression b2 4ac in
the general formula is called the discriminant of the equation, as
it determines the type of roots that the equation has.
e.g. 1: Find the range of values of k for which the equation 2x
2 + 5x + 3 k = 0 has two real distinct roots.
2
2
4 0(5) 4(2)(3 ) 025 24 8 01 8 08 1
18
b ack
kk
k
k
> > + >
+ >> >
e.g. 2: The roots of 3x 2 + kx +12 = 0 are equal. Find k.
2
2
2
2
4 0( ) 4(3)(12) 0
144 0144
14412
b ack
kk
kk
= =
=== =
e.g. 3: Find the range of values of p for which the equation x 2
2px + p2 + 5 p 6 = 0 has no real roots.
2
2 2
2 2
4 0( 2 ) 4(1)( 5 6) 04 4 20 24 0
20 24 020 24
20 24242065
b acp p p
p p ppp
p
p
p
< + >
>
b2 4ac > 0 two real and distinct roots b2 4ac = 0 two real
and equal roots b2 4ac < 0 no real roots b2 4ac 0 the roots are
real
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Quadratic Equation
9
e.g. 4: Show that the equation a 2 x 2 + 3ax + 2 = 0 always has
real roots.
2
2 2
2 2
2
4(3 ) 4( )(2)9 8
b aca a
a aa
= = =
a2 >0 for all values of a. Therefore
2 4 0b ac > Proven that a 2 x 2 + 3ax + 2 = 0 always has real
roots.
