Real Number System

Positive integers are also called natural numbers. Positive integers, negative integers and zero together form the system of integers. Integers Negative integers …, –4, –3, –2, –1, Zero 0, Positive integers 1, 2, 3, 4, … Positive integers are also called natural numbers.

p and q are integers and q ≠ 0 Rational numbers Rational numbers are numbers which can be expressed as . p q p and q are integers and q ≠ 0 Fractions and integers together form the system of rational numbers. For example: , , , 1 5 2 –9 17 4 3 1 1 –6 1 3 = , 0 = , –6 = Integers These are fractions. Rational numbers These are integers. Fractions

By long division, all fractions can be converted into terminating decimals or recurring decimals. Consider and . (i) (ii) ∴ ∴

Can all terminating decimals and recurring decimals be converted into fractions? For terminating decimal, (i) (ii)

All terminating decimals and recurring decimals are rational numbers. For recurring decimal, Let i.e. (2)  (1): All terminating decimals and recurring decimals are rational numbers. ∴

p and q are integers and q ≠ 0 Irrational numbers Irrational numbers are numbers which cannot be expressed as . p q p and q are integers and q ≠ 0 They can only be written as non-terminating and non-recurring decimals. For example:

Real numbers Rational and irrational numbers together form the real number system. Real numbers Rational numbers Irrational numbers ,  –4, When plotting each real number above on a real number line, we have

Now, we summarize the relationships among different kinds of real numbers as follows:

Follow-up question Consider the following numbers. , 8, , , 3.5, 4 , 8, , , 3.5, 4 Rational numbers Integer 8 Fraction Terminating decimal 3.5 Recurring decimal Irrational numbers , 4

Quadratic Equations in One Unknown

highest degree of the unknown is 2 What is quadratic equation in one unknown? x 2  5x + 6 = 0, 2 8x 2 + 2x  3 = 0, 2 highest degree of the unknown is 2 one unknown x 4y 2 = y + 1, 2 5y 2 = 20 2 one unknown y These equations are all quadratic equations in one unknown.

a, b and c are real numbers General form of quadratic equations in one unknown ax2 + bx + c = 0 a  0 a, b and c are real numbers If a = 0, the equation becomes a linear equation bx + c = 0. Rearranging terms For example: (i) 2x2  5x + 6 = 0 2  5 + 6 (ii) 2x  3 + x2 = 0 We usually keep a positive. x2 + 2x  3 = 0 1 + 2  3 a b c a b c

In fact, all quadratic equations can be written in general form. Rewrite the following quadratic equations in general form. By transposing terms, 2x2 + 5x  3 = 0 2x2 + 5x = 3 By expanding the equation, x2 + x + 3 = 0 x(x + 1) + 3 = 0 By expanding and transposing terms, x2  4x + 3 = 0 (x  2)2 = 1

What is a root of a quadratic equation ax2 + bx + c = 0? Roots of a quadratic equation What is a root of a quadratic equation ax2 + bx + c = 0? A root of an equation is a value of x that satisfies the equation. For example, 3 is a root of the equation x2  9 = 0.  32 – 9 = 0

Is 1 a root of the equation x 2  5x  6 = 0? Follow-up question Is 1 a root of the equation x 2  5x  6 = 0? Substitute x = 1 into the equation. L.H.S. = (1)2  5(1)  6 = 0 = R.H.S.  1 is a root of the equation x 2  5x  6 = 0.

Solving Quadratic Equations by Factor Method

For any expression in the form (px  r)(qx  s), For any two real numbers a and b, if (px  r)(qx  s) = 0 if ab = 0 a = 0 or b = 0 px  r = 0 or qx  s = 0

How to solve a quadratic equation using the factor method? ax2 + bx + c = 0 (px  r)(qx  s) = 0 factorize ax2 + bx + c px  r = 0 or qx  s = 0 x = or x = r p s q The roots of ax2 + bx + c = 0 are and . r p s q

x2 + x  12 = 0 using the factor method? Can you solve x2 + x  12 = 0 using the factor method? x2 + x  12 = 0 Factorize x2 + x  12 first. (x  3)(x + 4) = 0 x –3 +4 –3x +4x = +x x  3 = 0 or x + 4 = 0 x = 3 or x = 4  The roots of x2 + x  12 = 0 are 3 and 4.

 The roots of 2x2  x  6 = 0 are and 2. Follow-up question Solve 2x2  x  6 = 0 using the factor method. 2x2  x  6 = 0 Factorize 2x2 – x  6. (2x + 3)(x  2) = 0 2x + 3 = 0 or x  2 = 0 3 x =  2 or x = 2  The roots of 2x2  x  6 = 0 are and 2. 3 2

Solving Quadratic Equations by the Quadratic Formula

Method of taking square roots In fact, for quadratic equations in the form (x + m)2 = n, we can solve them by taking square roots. We have learnt how to solve quadratic equations ax2 + bx + c = 0 by the factor method. e.g. (x – 3)2 = 16

Follow-up question How to solve (x + 5)2 = –9? (x + 5)2 = –9 ∴ The equation has no real roots. is not a real number. Follow-up question Solve (x + 1)2 = 25 by taking square roots. (x + 1)2 = 25

Yes. There is a formula which can solve all quadratic equations. x2 + 3x – 2 cannot be factorized by cross-method. Is there any way to solve x2 + 3x – 2 = 0? Yes. There is a formula which can solve all quadratic equations.

Rewrite the equation in the form (x + m)2 = n. ax2 + bx + c = 0 , where a  0 Step 1 Rewrite the equation in the form (x + m)2 = n. Divide both sides by a. Add the term to both sides. 2 ç è æ a b

Step 2 Take square roots on both sides. Step 3 Express the roots in terms of a, b and c. Quadratic formula

Using the quadratic formula Now, I can solve x2 + 3x – 2 = 0. a = 1 c = –2 ∴ b = 3

Solve 2x2  4x + 3 = 0 using the quadratic formula. Follow-up question Solve 2x2  4x + 3 = 0 using the quadratic formula.  Substitute a = 2, b = –4 and c = 3 into the formula. ∵ is not a real number. ∴ The equation has no real roots.

Solving Quadratic Equations by the Graphical Method

Miss Chan, how to plot a quadratic graph like y = 2x2 + 5x  7? You can choose a range of values of x and find the corresponding value of y.

Consider the graph of y = 2x2 + 5x  7. Step 1 Find the value of y corresponding to each integral value of x from x = 4 to x = 1. x 4 3 2 1 1 y 5 4 9 10 7 Step 2 Plot the point with coordinates (x, y) for each pair of x and y in the table.

Join these points with a smooth curve. Step 3 y x-intercept 4 x-intercept Join these points with a smooth curve. 2 (–3.5, 0.0) (1.0, 0.0) x 3 2 1 The corresponding value of y is 0.0 when x equals –3.5 and 1.0. 4 1 2 4 y = 2x2 + 5x  7 6 8 parabolic shape 10

x-intercepts of y = 2x2 + 5x  7 are 3.5 and 1.0 2x2 + 5x  7 = 0 when x = 3.5 or 1.0 3.5 and 1.0 are the roots of 2x2 + 5x  7 = 0 The x-intercepts of the graph of y = ax2 + bx + c (where a ≠ 0) are the roots of ax2 + bx + c = 0.

When finding the roots of ax2 + bx + c = 0 by the graphical method, there are three possible cases: No. of x-intercepts of the graph of y = ax2 + bx + c The graph has two x-intercepts. e.g. The graph has only one x-intercept. e.g. The graph has no x-intercepts. e.g. y x y x y x Nature of roots of ax2 + bx + c = 0 2 unequal real roots / 2 distinct real roots It refers to whether the roots are 1. real or not real, 2. equal or unequal. 2 equal real roots / 1 double real root no real roots

Follow-up question Find the roots of the quadratic equation x2  2x + 3 = 0. The x-intercepts are 3 and 1. y 4 2 2 4 Therefore, the roots of x2  2x + 3 = 0 are 3 and 1. y = x2  2x + 3 x 4 3 2 1 1 2 Note: We can only find the approximate values of the roots by the graphical method.

Problems Leading to Quadratic Equations

I have learnt different methods of solving quadratic equations. How to use these methods to solve practical problems leading to quadratic equations? I have learnt different methods of solving quadratic equations. 1. Factor method 2. Quadratic formula 3. Graphical method Let’s see the following example.

The sum of a positive number and its square is 72. Find the number. Let x be the number.  Step 1: Identify the unknown quantity and use a letter, say x, to represent it. ∴ x + x2 = 72  Step 2: Form a quadratic equation according to the given conditions. x2 + x – 72 = 0 (x – 8)(x + 9) = 0  Step 3: Solve the equation using the factor method. x – 8 = 0 or x + 9 = 0 x = 8 or x = –9 (rejected)  Step 4: Check whether the solutions are reasonable. x must be positive ∴ The number is 8.

Follow-up question Mr Chan is 30 years older than his daughter. The product of their ages is 675. Find the age of Mr Chan. Let x be the age of Mr Chan, then x – 30 is the age of his daughter.  Represent the other unknown quantity in terms of x. ∴ x(x – 30) = 675 x2 – 30x – 675 = 0

By the quadratic formula Follow-up question Mr Chan is 30 years older than his daughter. The product of their ages is 675. Find the age of Mr Chan. By the quadratic formula  The age must be positive. ∴ Mr Chan is 45 years old.