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Chapter ( 2 )
Simultaneous equations and Quadratics
2.1 Simultaneous Equations
Solving simultaneous equations is finding the points of intersection of
the two graphs given by these equations.
Substitution Method
Example : 𝑦 = 𝑥
𝑥2 + 𝑦2 = 8
Substitute 𝑦 = 𝑥 𝑖𝑛 𝑥 2 + 𝑦 2 = 8 .
𝑥2 + 𝑥2 = 8
2𝑥 2 = 8
𝑥2 = 4
𝑥 = ±2
𝑊ℎ𝑒𝑛 𝑥 = 2 , 𝑦 = 2 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 = −2 , 𝑦 = −2 .
Points of intersection are ( 2, 2 ) 𝑎𝑛𝑑 ( −2 , −2 ).
2.2 Maximum and Minimum Values of a Quadratic Function
General form of a quadratic function is y = ax 2 +bx+c , where a ≠ 0.
Graph of a quadratic function
The graph of a quadratic function is called a parabola .
If a is positive ( i.e, a > 0 ) , the parabola opens up.
If a is negative ( i.e, a < 0 ) , the parabola opens down.
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shape
shape
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Complete square form and Vertex
𝑏
𝑦 = 𝑎𝑥 2 + 𝑏 𝑥 + 𝑐 = 𝑎 ( 𝑥 2 + 𝑎 𝑥 ) + 𝑐
𝑏
2
𝑏
𝑏
2
= 𝑎 [ 𝑥 2 + 𝑎 𝑥 + (2𝑎) − (2𝑎) ] + 𝑐
𝑏
2
𝑏2
𝑏
2
𝑏2
= 𝑎[( 𝑥 + 2𝑎 ) − 4𝑎2 ] + c
= 𝑎 ( 𝑥 + 2𝑎 ) −
𝑏
2
= a ( 𝑥 + 2𝑎 ) −
4𝑎
4𝑎𝑐
+ 4𝑎
( 𝑏 2 − 4𝑎𝑐 )
4𝑎
𝒃
= 𝑎( 𝑥 − ℎ )2 + 𝑘 , where h = − 𝟐𝒂 and k = −
( 𝒃𝟐 − 𝟒𝒂𝒄)
Vertex of a parabola [ Turning point , Stationary point ]
( 1 ) Vertex from the complete square form
𝑦 = a( x − h )2 + k
The vertex of the parabola is ( ℎ, 𝑘 )
b
Vertex = ( h, k ) = ( − 2a , −
b2 − 4ac
4a
).
( 2 ) Factor form and Vertex
y = 𝑎𝑥 2 + 𝑏 x + c = a( x – p )( x – q )
x- coordinate of vertex = h =
Vertex = ( h , k ) = (
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𝑝+𝑞
𝑝+𝑞
2
2
𝑝+𝑞
, y- coordinate of vertex = k = f ( 2 )
𝑝+𝑞
, 𝑓 ( 2 ))
𝟒𝒂
.
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Maximum and and Minimum value of y = 𝐚𝐱 𝟐 + 𝐛 x + c and Vertex
( 1 ) If a > 0 , y = ax 2 + b x + c has the minimum value at the vertex.
𝑏
y = a( x − h )2 + k , a >
0
Axis of symmetry, x = h
Minimum value = 𝑘 = 𝑓 (− 2𝑎 )
Vertex ( h, k )
Minimum point
( 2 ) If a < 0 , y = ax 2 + b x + c has the maximum value at the vertex.
𝑏
Maximum value = 𝑘 = 𝑓 (− 2𝑎 )
Maximum point
Axis of symmetry, x = h
Vertex ( h, k )
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y = a( x − h )2 + k , a < 0
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2.3 Graphs Of 𝒚 = | 𝒇( 𝒙 ) | 𝒘𝒉𝒆𝒓𝒆 𝒇( 𝒙 ) 𝒊𝒔 𝒂 𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏.
𝑓( 𝑥 ) = ax 2 + bx + c , a ≠ 0
y = | f( x ) | = | ax 2 + bx + c |
1. First draw the graph of 𝑦 = 𝑓( 𝑥 ) .
2. Reflect in the x-axis the part of the curve 𝑦 = ax 2 + bx + c that is
below the x-axis .
𝑦
𝑦 = |𝑓(𝑥)|
𝑥
𝑂
𝑦 = 𝑓( 𝑥 )
𝑦
𝑦 = |𝑓(𝑥)|
𝑥
𝑂
𝑦 = 𝑓( 𝑥 )
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2.4 Quadratic Inequality
ax 2 + bx + c < 0
ax 2 + bx + c ≤ 0
ax 2 + bx + c > 0
ax 2 + bx + c ≥ 0
The solution set of an inequation ( y > 0 or y < 0 ) is the set of all x
which satisfies this inequation.
b2 − 4ac > 0
y = ax 2 + bx + c = a ( x – p ) ( x – q ) and p < q .
y = a(x − p)(x − q) ,a>0
y = a(x − p)(x − q),a<0
+
p
+
+
p
q
−
q
x
−
𝑦 > 0 → 𝑥 < 𝑝 𝑜𝑟 𝑥 > 𝑞
x
−
𝑦>0 →𝑝<𝑥< 𝑞
𝑦 < 0 → 𝑥 < 𝑝 𝑜𝑟 𝑥 > 𝑞
𝑦<0 →𝑝<𝑥<𝑞
2.5 Roots Of A quadratic Equation
The solutions of a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are called
the roots of this quadratic equation.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 ; 2𝑥 2 − 3𝑥 − 5 = 0
( 2𝑥 − 5) ( 𝑥 + 1 ) = 0
2𝑥 − 5 = 0 𝑜𝑟 𝑥 + 1 = 0
5
𝑥=2
𝑜𝑟 𝑥 = −1
5
𝑇ℎ𝑒 𝑟𝑜𝑜𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑒 − 1 𝑎𝑛𝑑 2 .
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Quadratic Formula of ax 2 + bx + c = 0 .
ax 2 + bx + c = 0
ax 2 + bx = − c
b
x
a
= −
c
a
b
b 2
x +( )
a
2a
= −
c
a
x2 +
x2 +
(x+
b 2
2a
+ ( )
b 2
4ac
b2
) = − 2 + 2
2a
4a
4a
b
2
x+
b
2a
b2 − 4ac
4a2
( x + 2a ) =
= ± √
x = −
x=
b
2a
b2 − 4ac
4a2
±
√ b2 − 4ac
2𝑎
−b ± √ b2 − 4ac
2a
Discriminant of a Quadratic Function
y = ax 2 + b x + c
𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 = b2 − 4ac .
( i ) If 𝑏 2 − 4𝑎𝑐 > 0 ,
ax 2 + b x + c = 0 has two distinct real roots and so
the graph cuts the x- axis twice.
y = ax 2 + bx + c , a < 0
y = ax 2 + bx + c , a > 0
b2 − 4ac > 0
b2 − 4ac > 0
x
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( ii ) If 𝑏 2 − 4𝑎𝑐 = 0 ,
ax 2 + b x + c = 0 has two equal real roots ( only one root ) and so
the graph is tangent to the x- axis at only one point.
y = ax 2 + bx + c , a < 0
2
y = ax + bx + c , a > 0
b2 − 4ac < 0
b2 − 4ac = 0
x
x
( iii ) If 𝑏 2 − 4𝑎𝑐 < 0 ,
ax 2 + b x + c = 0 has no real root and so
the graph does cut the x- axis .
y = ax 2 + bx + c , a < 0
y = ax 2 + bx + c , a > 0
b2 − 4ac < 0
b2 − 4ac < 0
x
x
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2.6
Intersection Of A Line And A Conic Section Curve
Conic Sections are Parabolas , Circles , Ellipses and Hyperbolas .
After substituting the line equation in conic section equation
( 𝐴 𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 ) , we will have the quadratic equation
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 .
Thus we can determine whether or not the line intersect the curve
as following:
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Discriminant
𝑏 2 − 4𝑎𝑐 > 0
Two points of
intersection
Discriminant
𝑏 2 − 4𝑎𝑐 = 0
Exactly one point of
Intersection
i.e. the line is tangent
to the curve.
Discriminant
𝑏 2 − 4𝑎𝑐 < 0
No point of
intersection
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A maths ( 0606 ) Quadratic and Modulus
2.
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