Download Quadratic Equations

Crescent Girls' School
Mathematics Department
Quadratic Equations
Contents
1.1
1.2
1.3
1.4
1.5
1.6
Introduction
Solve quadratic equation by completing the square
Solve quadratic equation by formula
Solve fractional equations
Solve word problems on quadratic equations
Summary
1.1
Introduction
What is a Quadratic Equation?
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Examples of quadratic equations are
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We have learnt how to solve quadratic equations in Lower Secondary using
Factorisation.
Example 1
a)
Solve the following equations by factorisation:
2 x 2  8x
b)
2x  1x  2  5
NOTE: Did you substitute your solutions back into the equation to confirm your answer?
Now try solving 8 x 2  9 x  2  0 using factorisation again.
Can it be done?
Is there another other methods we can use to solve the above equation?
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LBK/2003
Crescent Girls' School
1.2
Mathematics Department
Solve quadratic equations by completing the square
Let us try to solve the following equations
a)
 x  2 2
 25
x  32  8
b)
It can be deduced that any quadratic equation written in the form x  p   q , where p
and q are real numbers, the solutions can be found.
2
Note: Expressions of the form of a  b  or a  b  are referred as perfect squares.
E.g. _______________________________________________________________
2
2
Let us take a look at this equation 8 x 2  9 x  2  0 given earlier again. We will solve the
equations by following the steps below:
Step1: Divide both sides of the equation by the coefficient of x2 if it is not 1.
Step 2: Subtract the constant term from both sides.
Step 3: Add the square of half the coefficient of x to both sides.
Step 4: Rewrite the left-hand side as a perfect square.
Step 5: Solve the equation by taking square root on both sides.
This method is call " Completing the square" method.
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Crescent Girls' School
Example 2
Mathematics Department
Solve the following equations using "Completing the square" method,
giving your answer correct to 3 significant figures.
a)
2x 2  7x  2  0
b)
5x 2  4 x  2  0
c)
1 2 3
1
x  x 0
4
8
2
Homework (Practice 3A) Pg. 43
Q1(c, d, f, g, h), 2(c, f, g, j)
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LBK/2003
Crescent Girls' School
Mathematics Department
1.3 Solve quadratic equation by formula
We have learnt how to solve quadratic equations using the method of completing the
square. Since the general form of a quadratic equation is __________________, where a,
b, and c are real numbers and a  0 , we shall use the method of completing the square to
derive a formula for the solution to all quadratic equations.
ax 2  bx  c  0
Step1: Divide both sides of the equation by the coefficient of x2 if it is not 1.
Step 2: Subtract the constant term from both sides.
Step 3: Add the square of half the coefficient of x to both sides.
Step 4: Rewrite the left-hand side as a perfect square.
Step 5: Solve the equation by taking square root on both sides.
Hence, if ax 2  bx  c  0 , then x 
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 b  b 2  4ac
2a
LBK/2003
Crescent Girls' School
Example 3
Mathematics Department
Solve the following equations using the formula. Give your answers
corrected to 3 significant figures.
a)
14 x 2  17 x  6  0
b)
x2 3
2
 x 0
6 4
3
c)
x 6 4 x
  
6 x x 4
Homework (Practice 3B) Pg. 45
Q1(e, f), 2(a, d, k, t, x, y, z)
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LBK/2003
Crescent Girls' School
1.4
Mathematics Department
Solve fractional equations
We shall now learn about fractional equations which can be reduced to quadratic
equations.
Example 4
Express the following equations into the form ax 2  bx  c  0 .
a)
1
1
1


x x  3 60
c)
2
d 6
 2
0
d 3 d 9
d)
4
3
 2
0
m  3m  2 m  5m  6
b)
x
9x
5x  3

x3
12
2
Homework (Practice 3C) Pg. 47
Q1(b, d), 2(b, c), 3a, 4(b, d), 5b
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LBK/2003
Crescent Girls' School
Mathematics Department
1.5
Solve word problems on quadratic equations
Example 5
a)
The product of two consecutive odd numbers is 225. Find the numbers.
b)
A wire of length 200cm is cut into two parts and each is bent to form a square. If
the area of the larger square is 9 times that of the smaller square, find the
perimeter of the larger square.
c)
An aircraft flies from A to B, 450km away, and back from B to A in a total time
of 5.5 hours. During the whole journey there is a constant wind blowing from A
to B. If the speed of the aircraft in still air is 165 km/h, find the speed of the
wind.
d)
A group of students are on a sightseeing tour. The total fare is $120 and is to be
shared equally among the students. If two more students join in the tour, each
will pay $2 less. Find the original number of students in a group.
Homework (Practice 3D) Pg. 50
Q6, 7, 8, 11, 15, 16, 17, 20
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Crescent Girls' School
1.6
Mathematics Department
Summary
Solving Quadratic Equations
To solve quadratic equations, there are 3 methods, namely
1)
Factorisation
Solve the equation 3x2 – 2x – 8 = 0
3x2 – 2x – 8 = 0
(x-2)(3x +4) = 0
Therefore: x-2 = 0 or 3x + 4 = 0
x = 2 or 3x = -4
x = 2 or x = -1 1
3
2)
Completing the Square
2x2 +x – 6 = 0
x
x2 + - 3 = 0
2
1
x
1
x2 + + ( ) 2 = 3 + ( ) 2
2
4
4
49
1
(x + )2 =
4
16
1
49
x+ = 
4
16
x = 1.5 or -2
3)
Formula
The General quadratic equation is ax2 + bx + c = 0
Its solution is given by the formula,
x=
 b  b 2 - 4ac
2a
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