Exercise 1

Determine the quadratic equation whose solutions are 3 and −2.

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Exercise 2

Factor:

x^2 - 5x + 6 = 0

Exercise 3

Determine the value of k so that the two roots of the equation x² − kx + 36 = 0 are equal.

Exercise 4

The sum of two numbers is 5 and their product is −84. Find these numbers.

Exercise 5

Within 11 years, the age of Peter will be half the square of the age he was 13 years ago. Calculate the current age of Peter.

Exercise 6

To fence a rectangular farm of 750 m², 110 m of fence has been used. Calculate the dimensions of the farm.

Exercise 7

The three sides of a right-angled triangle are proportional to the numbers 3, 4 and 5. Find the length of each side knowing that the area of the triangle is 24 m².

Exercise 8

A rectangular garden 50 m long and 34 m wide is surrounded by a uniform dirt road. Find the width of the road if the total area of the garden and road is 540 m².

Exercise 9

Calculate the dimensions of a rectangle whose diagonal is 75 m, knowing that is similar to a rectangle with sides measuring 36 m 48 m respectively.

Exercise 10

Two natural numbers differ by two units and the sum of their squares is 580. What are these numbers?

Exercise 11

Two taps A and B fill a swimming pool together in two hours. Alone, it takes tap A three hours less than B to fill the same pool. How many hours does it take each tap to fill the pool separately?

Exercise 12

The length of the sides of a right-angled triangle are measured as three consecutive even numbers (in cm). Find the values of these sides.

Exercise 13

A rectangular piece of cardboard is 4 cm longer than wide. A box of 840 cm³ is constructed by using this piece of cardboard. A square of 6 cm is cut out in every corner and the rims are folded upwards to create the box. Find the dimensions of the box.

Exercise 14

2 faucets can fill a tank in 1 hour and 20 minutes. The first faucet takes more than two hours longer to fill the same tank when functioning without the second tap. How long does it take to fill each one separately?

 

 

Solution of exercise 1

Determine the quadratic equation whose solutions are: 3 and −2.

S = 3 − 2 = 1

P = 3 · 2 = 6

x² − x + 6 = 0

 

Solution of exercise 2

Factor:

x^2 - 5x + 6 = 0

x = \frac{5 \pm \sqrt{5^2 - 4 \cdot 6}}{2}

= \frac{5 \pm \sqrt{25 - 24}} {2}

= \frac{5 \pm 1}{2}

x_1 = \frac{6}{2} = 3

x_2 = \frac{4}{2} = 2

(x - 2) (x - 3) = 0

 

Solution of exercise 3

Determine the value of k so that the two roots of the equation x² − kx + 36 = 0 are equal.

b^2 - 4ac = 0

k^2 - 4 \cdot 36 = 0

k^2 = 144

k = \pm\sqrt{144}

k_1 = 12

k_2 = -12

b^2 - 4ac = 0

k^2 - 4 \cdot 36 = 0

k^2 = 144

 

Solution of exercise 4

The sum of two numbers is 5 and their product is −84. Find these numbers.

x^2 - Sx + P = 0

x^2 - 5x - 84 = 0

x = \frac{5 \pm \sqrt{25 + 336}} {2}

= \frac{5 \pm \sqrt{361}}{2}

= \frac{5 \pm 19}{2}

x_1 = \frac{24}{2} = 12

x_2 = \frac{-14}{2} = -7

Solution of exercise 5

Within 11 years, the age of Peter will be half the square of the age he was 13 years ago. Calculate the current age of Peter.

Current age x

Age 13 years ago x − 13

Age within 11 years x + 11

x + 11 = \frac{(x - 13)^2}{2}

2x + 22 = x^2 - 26x + 169

x^2 - 28x + 147 = 0

x = 21

Current age 21

 

Solution of exercise 6

To fence a rectangular farm of 750 m², 110 m of fence has been used. Calculate the dimensions of the farm.

Exercise 6 - Solution

Semiperimeter 55

Base x

Height 55 − x

x · (55 − x) = 750

x² − 55x + 750 = 0

x = 25      x = 30

The dimensions of the farm are 30 m and 25 m.

Solution of exercise 7

The three sides of a right-angled triangle are proportional to the numbers 3, 4 and 5. Find the length of each side knowing that the area of the triangle is 24 m².

Exercise 7 - Solution

1st side (base) 3x

2nd side (height) 4x

3rd side 5x

\frac{3x \cdot 4x}{2} = 24

x^2 = 4

x = \pm 2

x = 2

1st side 6 m

2nd side 8 m

3rd side 10 m

 

Solution of exercise 8

A rectangular garden 50 m long and 34 m wide is surrounded by a uniform dirt road. Find the width of the road if the total area of the garden and road is 540 m².

Exercise 8 - Solution

(50 + 2x) · (34 + 2x) − 50 · 34 = 540

4x² + 168x − 540 = 0        x² + 42x − 135 = 0

x = 3 and x = −45

The road width is 3 m .

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Solution of exercise 9

Calculate the dimensions of a rectangle whose diagonal is 75 m, knowing that it is similar to a rectangle with sides measuring 36 m 48 m respectively.

Base 48x : 12 = 4x

Height 36x : 12 = 3x

Exercise 9 - Solution

(4x)² + (3x)² = 75²

25x² = 5625

x² = 225      x = 15

Base 4 · 15 = 60 m

Height 3 · 15 = 45 m

 

Solution of exercise 10

Two natural numbers differ by two units and the sum of their squares is 580. What are these numbers?

1st number x

2nd number x + 2

x^2 + (x + 2)^2 = 580

x^2 + x^2 + 4x + 4 = 580

2x^2 + 4x - 576 = 0

x^2 + 2x - 288 = 0

x = \frac{-2 \pm \sqrt{4 + 1152}} {2}

= \frac{-2 \pm 34}{2}

x_1 = 16

x_2 = -18

1st number 16

2nd number 18

 

Solution of exercise 11

Two taps A and B fill a swimming pool together in two hours. Alone, it takes tap A three hours less than B to fill the same pool. How many hours does it take each tap to fill the pool separately?

Time of A x

Time of B x + 3

A

B

A and B

\frac{1}{x} + \frac{1}{x + 3} = \frac{1}{2}

LCM(2, x, x + 3) = 2x (x + 3)

2x + 6 + 2x = x^2 + 3x              x^2 - x - 6 = 0

x = \frac{1 \pm \sqrt{1 + 24}}{2}

x_1 = 3

x_2 = -2

Time of A 3 hours

Time of B 6 hours

 

Solution of exercise 12

The length of the sides of a right-angled triangle are measured as three consecutive even numbers (in cm). Find the values of these sides.

Exercise 12 - Solution

1st leg 2x

2nd leg 2x + 2

Hypotenuse 2x + 4

(2x)² + (2x + 2)² = (2x + 4)²

4x² + 4x² + 8x + 4 = 4x² + 16x + 16

4x² − 8x − 12 = 0         x² − 2x − 3 = 0

x = 3 y x= −1

1st leg 6 cm

2nd leg 8 cm

Hypotenuse 10 cm

 

Solution of exercise 13

A rectangular piece of cardboard is 4 cm longer than wide. A box of 840 cm³ is constructed by using this piece of cardboard. A square of 6 cm is cut out in every corner and the rims are folded upwards to create the box. Find the dimensions of the box.

Solution of Exercise 13

6 (x − 12) · (x + 4 −12) = 840       (x − 12) · (x −8) = 140

x² − 20x − 44 = 0    x = 22 y x= −2

The dimensions are: 26 cm and 22 cm.

 

Solution of exercise 14

2 faucets can fill a tank in 1 hour and 20 minutes. The first faucet takes more than two hours longer to fill the same tank when functioning without the second tap. How long does it take to fill each one separately?

1st Time x

2nd Time x − 2

Between the two

\frac{1}{x} + \frac{1}{x - 2} = \frac{1} {\frac{4}{3}}

3x^2 - 14x + 8 = 0

x = \frac{14 \pm \sqrt{196 - 96}} {6}

= \frac{14 \pm 10}{6}

x_1 = 4

x_2 = \frac{2}{3}

1st Time 4 hours

2nd Time 2 hours

\frac{2}{3} is not a solution because the time for the second faucet would be negative.

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.