LemonmathSelf-study sheet

Grade 9 · Lesson 6.2

Quadratic equations in one unknown

Name: Date: Time spent: min

Goals

Recognise a quadratic equation in one unknown and identify its coefficients $a$ , $b$ , $c$ .
Solve special quadratic equations ( $b=0$ or $c=0$ ) and understand completing the square.
Solve quadratic equations with the quadratic formula and with the reduced discriminant.
Apply quadratic equations to real-life problems.

Remember

A quadratic equation: $ax^2+bx+c=0$ with $a\neq0$ .
$c=0$ : take out the common factor $x$ ; $b=0$ : rewrite as $x^2=k$ .
$\Delta=b^2-4ac$ : $\Delta>0$ two roots $x_{1,2}=\dfrac{-b\pm\sqrt\Delta}{2a}$ ; $\Delta=0$ a repeated root $-\dfrac{b}{2a}$ ; $\Delta<0$ no solutions.
$b=2b'$ : $\Delta'=b'^2-ac$ , $x_{1,2}=\dfrac{-b'\pm\sqrt{\Delta'}}{a}$ .
If $a$ and $c$ have opposite signs, there are always two distinct roots.
Real-life problems: check each root against the condition on the unknown.

Exercises

Exercise 1

Solve the following equations:

a) $5x^2-20=0$ .

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

Exercise 2

Solve the following equations using the quadratic formula:

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

b) $2x^2+x-1=0$ .

c) $9x^2-6x+1=0$ .

Exercise 3

Solve the following equations using the reduced discriminant:

a) $x^2-4x-1=0$ .

b) $5x^2+2x+1=0$ .

Exercise 4

Solve the following equations:

a) $(x+1)(x-3)=2x-7$ .

b) $x(x+2)=3(x+4)$ .

Exercise 5

Suppose a ball is kicked straight up from the ground and its height after $t$ seconds is $h=-5t^2+20t$ (metres). After how many seconds is the ball at a height of $15$ m?

Exercise 6

Suppose Mrs Lan puts $100$ million dong into a savings account at an interest rate $r$ per year, and the first year's interest is added to the capital before the second year's interest is worked out (compound interest). After $2$ years she receives $112.36$ million dong. Find the interest rate $r$ .

Exercise 7

Solve the equation $\dfrac{x}{x-2}-\dfrac1x=\dfrac{2}{x(x-2)}$ . (Revise equations with the unknown in a denominator.)

Exercise 8

(2.0 marks)

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

b) Find the coordinates of the points where the parabola $(P): y=x^2$ meets the line $(d): y=x+2$ .

c) Solve the equation $x^2-2\sqrt2x-6=0$ .

Exercise 9 Challenge

From a rectangular metal sheet $30$ cm long and $20$ cm wide, four equal squares are cut from the corners and the sides are folded up to make an open box. How long must the side of each square be for the base of the box to have area $336$ cm²? What is the volume of the box then, in cubic centimetres?

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. Which of the following is a quadratic equation in one unknown?

2x+3=0

.B.

x^3-x=0

.C.

-x^2+5=0

.D.

0x^2+3x-1=0

Question 2. The discriminant

\Delta

of the equation

2x^2-3x-2=0

25

.B.

-7

.C.

17

.D.

7

Question 3. The equation

x^2-6x+9=0

A. has two distinct roots

3

and

-3

.B. has a repeated root

x=3

.C. has a repeated root

x=-3

.D. has no solutions.

Question 4. The reduced discriminant

\Delta'

of the equation

3x^2-4x-1=0

7

.B.

1

.C.

28

.D.

19

Part II. True or false

Question 1. Consider the equation

x^2-2x-8=0

. Decide whether each statement is true or false.

a) The coefficients are $a=1$ , $b=-2$ , $c=-8$ .	T / F
b) $\Delta'=9$ .	T / F
c) The equation has the two roots $x=-4$ and $x=2$ .	T / F
d) The equation has a repeated root.	T / F

Part III. Short answer

Question 1. Find the positive root of the equation

x^2+x-12=0

Question 2. Find the smaller root of the equation

2x^2-5x-3=0

(give your answer as a decimal).

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

Hints (only when stuck)

Exercise 1.

What you need: if $b=0$ , rewrite as $x^2=k$ ; if $c=0$ , take out a common factor.

Steps:

a) Rearrange and divide by $5$ .
b) Write $-2x(x-3)=0$ .

Exercise 2.

What you need: $\Delta=b^2-4ac$ and the quadratic formula.

Steps:

Identify $a$ , $b$ , $c$ and work out $\Delta$ .
Look at the sign of $\Delta$ , then apply the formula.

Exercise 3.

What you need: $b=2b'$ , $\Delta'=b'^2-ac$ .

Steps:

Find $b'$ and work out $\Delta'$ .
Apply $x_{1,2}=\dfrac{-b'\pm\sqrt{\Delta'}}{a}$ .

Exercise 4.

What you need: expand and rearrange into the form $ax^2+bx+c=0$ .

Steps:

Expand both sides and move everything to the left-hand side.
Solve the resulting quadratic equation.

Exercise 5.

What you need: solve the equation $-5t^2+20t=15$ with $t\geq0$ .

Steps:

Rearrange and divide both sides by $-5$ .
Solve the resulting equation, then explain what the two roots mean.

Exercise 6.

What you need: each year the amount is multiplied by $1+r$ .

Steps:

Form the equation $100(1+r)^2=112.36$ .
Solve it as $X^2=k$ with $X=1+r>0$ .

Exercise 7.

What you need: find the conditions for the equation to be defined, clear the denominators and check the conditions.

Steps:

Conditions: $x\neq0$ , $x\neq2$ .
Multiply both sides by $x(x-2)$ .
Solve the resulting quadratic equation and reject any root that breaks the conditions.

Exercise 8.

What you need: the quadratic formula; the $x$ -coordinates of the points where $(P)$ meets $(d)$ are the roots of $x^2=x+2$ .

Steps:

a) Work out $\Delta$ .
b) Solve $x^2-x-2=0$ , then work out $y=x^2$ .
c) Use $b'=-\sqrt2$ .

Exercise 9.

What you need: the base of the box is a rectangle $(30-2x)\times(20-2x)$ , and the height of the box is $x$ .

Steps:

Let the side of the square be $x$ (cm), $0<x<10$ .
Form and solve the equation $(30-2x)(20-2x)=336$ .
Check against the condition, then find the volume.

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