LemonmathSelf-study sheet

Grade 9 · Lesson 6.3

Viète's theorem and its applications

Name: Date: Time spent: min

Goals

Understand and explain Viète's theorem on the sum and product of the roots of a quadratic equation.
Find the roots mentally when $a+b+c=0$ or $a-b+c=0$ .
Find two numbers given their sum and product.
Evaluate symmetric expressions in the two roots and find the signs of the roots without solving the equation.

Remember

If $\Delta\geq0$ : $x_1+x_2=-\dfrac ba$ , $x_1x_2=\dfrac ca$ .
$a+b+c=0$ : roots $1$ and $\dfrac ca$ ; $a-b+c=0$ : roots $-1$ and $-\dfrac ca$ .
Two numbers with sum $S$ and product $P$ are the roots of $x^2-Sx+P=0$ (need $S^2-4P\geq0$ ).
$x_1^2+x_2^2=S^2-2P$ ; $\dfrac1{x_1}+\dfrac1{x_2}=\dfrac SP$ .
$P<0$ : opposite signs; $P>0$ , $S>0$ : both positive; $P>0$ , $S<0$ : both negative (when $\Delta\geq0$ ).

Exercises

Exercise 1

Without solving, find the sum and the product of the roots (if any) of each equation:

a) $x^2-6x+2=0$ .

b) $3x^2+5x-4=0$ .

c) $x^2+x+1=0$ .

Exercise 2

Find the roots of the following equations mentally:

a) $5x^2-3x-2=0$ .

b) $x^2+9x+8=0$ .

c) $7x^2+4x-3=0$ .

Exercise 3

The equation $x^2-4x+1=0$ has the root $x_1=2+\sqrt3$ . Find the other root. (Revise rationalising the denominator.)

Exercise 4

Find the two numbers $u$ , $v$ in each case:

a) $u+v=10$ , $uv=21$ .

b) $u+v=-3$ , $uv=-40$ .

c) $u+v=6$ , $uv=10$ .

Exercise 5

Let $x_1$ , $x_2$ be the roots of the equation $x^2-3x-5=0$ . Without solving the equation, find:

a) $x_1^2+x_2^2$ .

b) $\dfrac1{x_1}+\dfrac1{x_2}$ .

c) $x_1^2x_2+x_1x_2^2$ .

Exercise 6

Without solving, find the signs of the roots of each equation:

a) $2x^2+5x-9=0$ .

b) $x^2-10x+7=0$ .

c) $x^2+8x+3=0$ .

Exercise 7

Suppose a rectangular outdoor sports court with spectator area has perimeter $120$ m and area $875$ m². Find the length and the width of the pitch.

Exercise 8

(1.5 marks) Consider the equation $x^2-5x+2=0$ .

a) Prove that the equation has two distinct roots $x_1$ , $x_2$ .

b) Without solving the equation, find $x_1^2+x_2^2$ .

c) Find the value of the expression $A=\dfrac{x_1}{x_2}+\dfrac{x_2}{x_1}$ .

Exercise 9 Challenge

Let $x_1$ , $x_2$ be the roots of the equation $x^2-6x+4=0$ . Prove that both roots are positive, then find $\sqrt{x_1}+\sqrt{x_2}$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The sum of the roots of the equation

2x^2-6x-5=0

3

.B.

-3

.C.

-\dfrac52

.D.

6

Question 2. The product of the roots of the equation

3x^2+7x-6=0

-2

.B.

2

.C.

-\dfrac73

.D.

-6

Question 3. The roots of the equation

4x^2-9x+5=0

are

1

and

\dfrac54

.B.

-1

and

-\dfrac54

.C.

1

and

-\dfrac54

.D.

1

and

5

Question 4. Two numbers with sum

9

and product

20

are the roots of the equation

x^2-9x+20=0

.B.

x^2+9x+20=0

.C.

x^2-20x+9=0

.D.

x^2+9x-20=0

Part II. True or false

Question 1. The equation

x^2-3x-10=0

has two roots

x_1

x_2

. Decide whether each statement is true or false.

a) The equation has two roots of opposite signs.	T / F
b) $x_1+x_2=-3$ .	T / F
c) $x_1x_2=-10$ .	T / F
d) $x_1^2+x_2^2=29$ .	T / F

Part III. Short answer

Question 1. Let

x_1

x_2

be the roots of the equation

x^2-7x+2=0

. Find

x_1^2+x_2^2

Question 2. Find mentally the larger root of the equation

2x^2-11x+9=0

(give your answer as a decimal).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: Viète's theorem, used only when the equation has roots.

Steps:

Check that $\Delta\geq0$ or that $a$ and $c$ have opposite signs.
Work out $-\dfrac ba$ and $\dfrac ca$ .

Exercise 2.

What you need: check $a+b+c$ or $a-b+c$ .

Steps:

Work out $a+b+c$ ; if it is not $0$ , work out $a-b+c$ .
Write the other root $\dfrac ca$ or $-\dfrac ca$ .

Exercise 3.

What you need: $x_1+x_2=4$ (or $x_1x_2=1$ ).

Steps:

Work out $x_2=4-x_1$ .
Check with the product: $x_2=\dfrac1{2+\sqrt3}$ .

Exercise 4.

What you need: $u$ , $v$ are the roots of $x^2-Sx+P=0$ , which needs $S^2-4P\geq0$ .

Steps:

Form the equation $x^2-Sx+P=0$ .
Solve it (or show that it has no solutions).

Exercise 5.

What you need: $S=x_1+x_2$ , $P=x_1x_2$ ; write each expression in terms of $S$ and $P$ .

Steps:

Check that the equation has two roots and find $S$ , $P$ .
Use $S^2-2P$ , $\dfrac SP$ and $x_1x_2(x_1+x_2)=PS$ .

Exercise 6.

What you need: $P<0$ : opposite signs; $\Delta\geq0$ and $P>0$ : both have the sign of $S$ .

Steps:

Work out $P$ ; if $P>0$ , check $\Delta'$ and work out $S$ .

Exercise 7.

What you need: the two sides have sum $S=60$ and product $P=875$ .

Steps:

Form the equation $x^2-60x+875=0$ .
Solve it with the reduced discriminant.

Exercise 8.

What you need: $\Delta>0$ ; $x_1^2+x_2^2=S^2-2P$ ; over a common denominator $A=\dfrac{x_1^2+x_2^2}{x_1x_2}$ .

Steps:

Work out $\Delta$ .
Find $S$ , $P$ , then $S^2-2P$ .
Use the result of part b) in part c).

Exercise 9.

What you need: the signs of the roots; $(\sqrt{x_1}+\sqrt{x_2})^2=x_1+x_2+2\sqrt{x_1x_2}$ .

Steps:

Work out $\Delta'$ , $S$ , $P$ .
Square the required expression and use $S$ , $P$ .
Take the square root, remembering that the expression is positive.

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