1. QUADRATICS
P1/1/1: Quadratic expressions of the form
ax2 + bx + c and their graphs
P1/1/2: Solving quadratic equation in one unknown
P1/1/1
P1/1/3: Nature of roots of quadratic expression
Quadratic expressions of the form
P1/1/4: Simultaneous equations of which one is ax2 + bx + c and their graphs
linear and one is quadratic
P1/1/5: Linear inequalities and quadratic inequalities
P1/1/6: Summary of lesson
Prepared by
2010-3-30 P1/1:QUADRATICS Tan Bee Hong
1
Quadratics expression
Learning Outcome
ax 2 + bx + c
where a(a ≠ 0), b and c are constants (coefficients).
Students should be able to:
The graph is a parabola.
• carry out the process of completing the square for a
quadratic polynomial. If a > 0 or
• locate the vertex of the quadratics graphs from the
completed form.
If a < 0
• sketch the quadratic graph
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Completed square form Completing the square
f ( x) = x 2 − 10 x + 21 can be written as In general,
quadratic expression x 2 + bx + c ⇒ completed square form
f ( x) = ( x − 7)( x − 3) factor form x-intercept?
key point
2
Vertex?  1  1 2
f ( x ) = ( x − 5) 2 − 4 completed square form Range of f(x)?  x + b  = x + bx + b
2
 2  4
2
 1  1
⇒ x 2 + bx =  x + b  − b 2
 2  4
2
Graph Both sides plus c: x 2 + bx + c =  x + 1 b  − 1 b 2 + c
 
 2  4
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CAMBRIDGE 'A' LEVELS 1

2. Example 1: Example 2:
Express x + 14 x + 50 in completed square form.
2
Express 2 x 2 + 12 x − 5 in completed square form.
Locate the vertex and the axis of symmetry of the Locate the vertex and the axis of symmetry of the
quadratic graph. Find the least or greatest value of quadratic graph. Find the least or greatest value of
the expression, and the value of x for which this the expression, and the value of x for which this
occurs. occurs.
Graph Graph
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Example 3: Example 4:
Express 12 x + x − 6 in completed square form, and
2
Express 3 − 7 x − 3 x in completed square form.
2
Locate the vertex and the axis of symmetry of the use your result to find the factors of 12 x 2 + x − 6 .
quadratic graph. Find the least or greatest value of
the expression, and the value of x for which this
occurs.
Graph Graph
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Example 5: Practice Exercise
Find the range of the function
f ( x ) = ( x − 1)( x − 2 ) Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)
Exercise 4A (Page 54)
Q3(d), Q5(e), Q6(e)(f), Q8(c)(f), Q9(c)
Graph
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CAMBRIDGE 'A' LEVELS 2

3. Learning Outcome
P1/1/2 Students should be able to:
Solving quadratic equation in • use an appropriate method to solve a given quadratic
one unknown •
equation.
solve equations which can be reduced to quadratic
equations.
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Solving Quadratic equation in one unknown Solving Quadratic equation in one unknown
Solving quadratic equations by: Solving quadratic equations by:
(i) Factorization (ii) Completing the square method
Example 6: Example 7:
x + 3x − 4 = 0
2
2x2 + 7x + 3 = 0
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Solving Quadratic equation in one unknown Example 8:
Solving quadratic equations by: Use the quadratic formula to solve the following
(iii) Quadratic formula equations. Leave your answers in surd form. If
The solution of ax 2 + bx + c = 0, where a ≠ 0, is
there is no solution, say so.
− b ± b 2 − 4 ac (a ) 2 x 2 − 3x − 4 = 0
x=
2a
(b ) 2 x 2 − 3x + 4 = 0
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CAMBRIDGE 'A' LEVELS 3

4. Equation which reduce to quadratic equations Equation which reduce to quadratic equations
Example 10:
Example 9:
Solve the equation x = 6− x
Solve the equation x 4 − 5 x 2 + 4 = 0
(a) by letting y stand for x
(b) by squaring both sides of the equation
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Practice Exercise
Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)
Exercise 4B (Page 58) P1/1/3
Q1(e)(g)(i) Nature of roots of
Exercise 4C (Page 61) quadratic expression
Q4(d)(f), Q5(f)(l), Q6(d)(e)
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Nature of roots of quadratic expression
Learning Outcome The discriminant b 2 − 4ac
ax 2 + bx + c = 0
− b ± b 2 − 4ac
⇒x=
Students should be able to: 2a
• evaluate the discriminant of a quadratic polynomial. (ii) If b2 – 4ac > 0, the equation ax2 + bx + c = 0 will have
• use the discriminant to determine the nature of the roots. two roots.
• relate the nature of roots to the quadratic graph.
(iii) If b2 – 4ac < 0, there will be no roots.
(iv) If b2 – 4ac = 0, there is one root only or a repeated root.
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CAMBRIDGE 'A' LEVELS 4

5. Example 11: Example 12:
What can you deduce from the values of discriminants of The equation 3x2 + 5x – k = 0 has two real roots. What can
the quadratics in the following equations? you deduce about the value of the constant k?
(a ) 2x2 − 7 x + 3 = 0
(b ) x 2 − 3x + 4 = 0
(c ) x2 + 2x +1 = 0
The equation 3x2 + 5x – k = 0 has two distinct real roots.
What can you deduce about the value of the constant k?
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Example 13: Example 14:
x2
The equation + kx + 9 = 0 has no root. Deduce as much The equation -3 + kx - 2x2 = 0 has a repeated root. Find the
as you can about the values of k? values of k. (exact fractions or surds)
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Practice Exercise
Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)
Exercise 4B (Page 58) P1/1/4
Q4(e)(f)(g), Q5(e)(h)(i) Simultaneous equations of which
one is linear and one is quadratic
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CAMBRIDGE 'A' LEVELS 5

6. Simultaneous equations of which one is linear and
Learning Outcome one is quadratic
Example 15:
Solve the simultaneous equations
Students should be able to: y = x 2 − 3 x − 8, y = x−3
• solve by substitution a pair of simultaneous equations
of which one is linear and one is quadratic.
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Simultaneous equations of which one is linear and Simultaneous equations of which one is linear and
one is quadratic one is quadratic
Example 16: Example 17:
Solve the simultaneous equations At how many points does the line 3y – x = 15 meet the
curve 4x2 + 9y2 = 36.
x + 4 xy − 3 y = − 27 ,
2 2
y = 2 x − 12
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Practice Exercise
Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)
Exercise 4C (Page 61) P1/1/5
Q2(h), Q3(d) Linear inequalities and
quadratic inequalities
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CAMBRIDGE 'A' LEVELS 6

7. Linear inequalities and quadratic inequalities
Learning Outcome
a>b⇔b<a } Strict inequalities
Students should be able to:
a≥b⇔b≤a } Weak inequalities
• solve linear inequalities
• solve quadratic inequalities
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Algebraic method Example 18:
If a > 0, then these statements are equivalent Solve the linear inequalities.
(i ) 3 x + 7 > −5
x 2 ≤ a 2 ⇔ −a ≤ x ≤ a (ii ) − 3 x ≥ − 12
x ≥ a ⇔ x ≤ −a or x ≥ a
2 2
(iii )
1
(8 x + 1) − 2(x − 3 ) > 10
3
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Example 19: Example 20:
Solve the quadratic inequalities by: Solve the quadratic inequalities by using the algebraic
(a) Graphical method method.
(b) Tabular method
(i ) 8 − 3 x − x 2 > 0
(i ) ( x − 4 )( x + 1) ≥ 0
(ii ) x 2 + 3 x − 5 > 0
(ii ) (3 − 4 x )(3 x + 4 ) > 0
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CAMBRIDGE 'A' LEVELS 7

8. Practice Exercise
Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)
Exercise 5A (Page 68)
Q3(g), Q4(h), 5(i), 6(g)
Exercise 5B (Page 71)
Q1(i)(k), Q3(b)(c)(d)
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CAMBRIDGE 'A' LEVELS 8