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2.3 Exercises

From Förberedande kurs i matematik 1

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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
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| style="border-bottom:1px solid #000" width="5px" |  
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{{Ej vald flik|[[2.3 Andragradsuttryck|Theory]]}}
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{{Not selected tab|[[2.3 Quadratic expressions|Theory]]}}
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{{Vald flik|[[2.3 Övningar|Exercises]]}}
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{{Selected tab|[[2.3 Exercises|Exercises]]}}
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|width="25%" | <math>x^2+5x+3</math>
|width="25%" | <math>x^2+5x+3</math>
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</div>{{#NAVCONTENT:Answer|Svar 2.3:1|Solution a|Lösning 2.3:1a|Solution b|Lösning 2.3:1b|Solution c|Lösning 2.3:1c|Solution d|Lösning 2.3:1d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:1|Solution a|Solution 2.3:1a|Solution b|Solution 2.3:1b|Solution c|Solution 2.3:1c|Solution d|Solution 2.3:1d}}
===Exercise 2.3:2===
===Exercise 2.3:2===
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|width="33%" | <math>3x^2-10x+8=0</math>
|width="33%" | <math>3x^2-10x+8=0</math>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 2.3:2|Solution a|Lösning 2.3:2a|Solution b|Lösning 2.3:2b|Solution c|Lösning 2.3:2c|Solution d|Lösning 2.3:2d|Solution e|Lösning 2.3:2e|Solution f|Lösning 2.3:2f}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:2|Solution a|Solution 2.3:2a|Solution b|Solution 2.3:2b|Solution c|Solution 2.3:2c|Solution d|Solution 2.3:2d|Solution e|Solution 2.3:2e|Solution f|Solution 2.3:2f}}
===Exercise 2.3:3===
===Exercise 2.3:3===
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|width="50%" | <math>x(x^2-2x)+x(2-x)=0</math>
|width="50%" | <math>x(x^2-2x)+x(2-x)=0</math>
|}
|}
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</div>{{#NAVCONTENT:Answer|Svar 2.3:3|Solution a|Lösning 2.3:3a|Solution b|Lösning 2.3:3b|Solution c|Lösning 2.3:3c|Solution d|Lösning 2.3:3d|Solution e|Lösning 2.3:3e|Solution f|Lösning 2.3:3f}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:3|Solution a|Solution 2.3:3a|Solution b|Solution 2.3:3b|Solution c|Solution 2.3:3c|Solution d|Solution 2.3:3d|Solution e|Solution 2.3:3e|Solution f|Solution 2.3:3f}}
===Exercise 2.3:4===
===Exercise 2.3:4===
<div class="ovning">
<div class="ovning">
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Determine a second-degree equation which has roots
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Find a second-degree equation which has roots
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{| width="100%" cellspacing="10px"
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|a)
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|width="100" | <math>3\ </math> and <math>\ \sqrt{3}</math>
|width="100" | <math>3\ </math> and <math>\ \sqrt{3}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 2.3:4|Solution a|Lösning 2.3:4a|Solution b|Lösning 2.3:4b|Solution c|Lösning 2.3:4c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:4|Solution a|Solution 2.3:4a|Solution b|Solution 2.3:4b|Solution c|Solution 2.3:4c}}
===Exercise 2.3:5===
===Exercise 2.3:5===
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{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | Determine a second-degree equation which only has <math>\,-7\,</math> as a root.
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|width="100%" | Find a second-degree equation which only has <math>\,-7\,</math> as a root.
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|b)
|b)
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|width="100" | The equation <math>\,x^2+4x+b=0\,</math> has one root at <math>\,x=1\,</math>. Determine the value of the constant <math>\,b\,</math>.
|width="100" | The equation <math>\,x^2+4x+b=0\,</math> has one root at <math>\,x=1\,</math>. Determine the value of the constant <math>\,b\,</math>.
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</div>{{#NAVCONTENT:Answer|Svar 2.3:5|Solution a|Lösning 2.3:5a|Solution b|Lösning 2.3:5b|Solution c|Lösning 2.3:5c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:5|Solution a|Solution 2.3:5a|Solution b|Solution 2.3:5b|Solution c|Solution 2.3:5c}}
===Exercise 2.3:6===
===Exercise 2.3:6===
<div class="ovning">
<div class="ovning">
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Determine the smallest value that the following polynomial can take
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Determine the smallest value that the following polynomials can take
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%" | <math>x^2-4x+2</math>
|width="33%" | <math>x^2-4x+2</math>
|c)
|c)
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|width="33%" | <math>x^2-5x+7</math>
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|width="33%" | <math>x^2-5x+7</math>.
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|}
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</div>{{#NAVCONTENT:Answer|Svar 2.3:6|Solution a|Lösning 2.3:6a|Solution b|Lösning 2.3:6b|Solution c|Lösning 2.3:6c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:6|Solution a|Solution 2.3:6a|Solution b|Solution 2.3:6b|Solution c|Solution 2.3:6c}}
===Exercise 2.3:7===
===Exercise 2.3:7===
<div class="ovning">
<div class="ovning">
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Determine the largest value that the following polynomials can take.
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Determine the largest value that the following polynomials can take
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%" | <math>-x^2+3x-4</math>
|width="33%" | <math>-x^2+3x-4</math>
|c)
|c)
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|width="33%" | <math>x^2+x+1</math>
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|width="33%" | <math>x^2+x+1</math>.
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</div>{{#NAVCONTENT:Answer|Svar 2.3:7|Solution a|Lösning 2.3:7a|Solution b|Lösning 2.3:7b|Solution c|Lösning 2.3:7c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:7|Solution a|Solution 2.3:7a|Solution b|Solution 2.3:7b|Solution c|Solution 2.3:7c}}
===Exercise 2.3:8===
===Exercise 2.3:8===
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|width="33%" | <math>f(x)=(x-1)^2+2</math>
|width="33%" | <math>f(x)=(x-1)^2+2</math>
|c)
|c)
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|width="33%" | <math>f(x)=x^2-6x+11</math>
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|width="33%" | <math>f(x)=x^2-6x+11</math>.
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</div>{{#NAVCONTENT:Answer|Svar 2.3:8|Solution a|Lösning 2.3:8a|Solution b|Lösning 2.3:8b|Solution c|Lösning 2.3:8c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:8|Solution a|Solution 2.3:8a|Solution b|Solution 2.3:8b|Solution c|Solution 2.3:8c}}
===Exercise 2.3:9===
===Exercise 2.3:9===
<div class="ovning">
<div class="ovning">
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Find all the points where the x-axis and the following curves intersect.
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Find all the points where the following curves intersect the <math>x</math>-axis.
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{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%" | <math>y=3x^2-12x+9</math>
|width="33%" | <math>y=3x^2-12x+9</math>
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</div>{{#NAVCONTENT:Answer|Svar 2.3:9|Solution a|Lösning 2.3:9a|Solution b|Lösning 2.3:9b|Solution c|Lösning 2.3:9c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:9|Solution a|Solution 2.3:9a|Solution b|Solution 2.3:9b|Solution c|Solution 2.3:9c}}
===Exercise 2.3:10===
===Exercise 2.3:10===
<div class="ovning">
<div class="ovning">
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In the ''xy''-plane, draw in all the points whose coordinates <math>\,(x,y)\,</math> satisfy
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In the ''xy''-plane, shade in the area whose coordinates <math>\,(x,y)\,</math> satisfy
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|a)
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</div>{{#NAVCONTENT:Answer|Svar 2.3:10|Solution a|Lösning 2.3:10a|Solution b|Lösning 2.3:10b|Solution c|Lösning 2.3:10c|Solution d|Lösning 2.3:10d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.3:10|Solution a|Solution 2.3:10a|Solution b|Solution 2.3:10b|Solution c|Solution 2.3:10c|Solution d|Solution 2.3:10d}}

Current revision

       Theory          Exercises      

Exercise 2.3:1

Complete the square of the expressions

a) x2−2x b) x2+2x−1 c) 5+2x−x2 d) x2+5x+3

Answer | Solution a | Solution b | Solution c | Solution d

Exercise 2.3:2

Solve the following second order equations by completing the square

a) x2−4x+3=0 b) y2+2y−15=0 c) y2+3y+4=0
d) 4x2−28x+13=0 e) 5x2+2x−3=0 f) 3x2−10x+8=0

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f

Exercise 2.3:3

Solve the following equations directly

a) x(x+3)=0 b) (x−3)(x+5)=0
c) 5(3x−2)(x+8)=0 d) x(x+3)−x(2x−9)=0
e) (x+3)(x−1)−(x+3)(2x−9)=0 f) x(x2−2x)+x(2−x)=0

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f

Exercise 2.3:4

Find a second-degree equation which has roots

a) −1  and  2
b) 1+3   and  1−3 
c) 3  and  3 

Answer | Solution a | Solution b | Solution c

Exercise 2.3:5

a) Find a second-degree equation which only has −7 as a root.
b) Determine a value of x which makes the expression 4x2−28x+48 negative.
c) The equation x2+4x+b=0 has one root at x=1. Determine the value of the constant b.

Answer | Solution a | Solution b | Solution c

Exercise 2.3:6

Determine the smallest value that the following polynomials can take

a) x2−2x+1 b) x2−4x+2 c) x2−5x+7.

Answer | Solution a | Solution b | Solution c


Exercise 2.3:7

Determine the largest value that the following polynomials can take

a) 1−x2 b) −x2+3x−4 c) x2+x+1.

Answer | Solution a | Solution b | Solution c

Exercise 2.3:8

Sketch the graph of the following functions

a) f(x)=x2+1 b) f(x)=(x−1)2+2 c) f(x)=x2−6x+11.

Answer | Solution a | Solution b | Solution c

Exercise 2.3:9

Find all the points where the following curves intersect the x-axis.

a) y=x2−1 b) y=x2−5x+6 c) y=3x2−12x+9

Answer | Solution a | Solution b | Solution c

Exercise 2.3:10

In the xy-plane, shade in the area whose coordinates (xy) satisfy

a) yx2  and  y1 b) y1−x2  and  x2y−3
c) 1xy2 d) x2yx

Answer | Solution a | Solution b | Solution c | Solution d

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