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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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{{Mall:Ej vald flik|[[2.3 Andragradsuttryck|Teori]]}}
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{{Not selected tab|[[2.3 Quadratic expressions|Theory]]}}
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{{Mall:Vald flik|[[2.3 Övningar|Övningar]]}}
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{{Selected tab|[[2.3 Exercises|Exercises]]}}
| style="border-bottom:1px solid #000" width="100%"|  
| style="border-bottom:1px solid #000" width="100%"|  
|}
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===Övning 2.3:1===
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===Exercise 2.3:1===
<div class="ovning">
<div class="ovning">
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Kvadratkomplettera f&ouml;ljande uttryck
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Complete the square of the expressions
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 20: Line 20:
|width="25%" | <math>x^2+5x+3</math>
|width="25%" | <math>x^2+5x+3</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.3:1|Lösning a|Lösning 2.3:1a|Lösning b|Lösning 2.3:1b|Lösning c|Lösning 2.3:1c|Lösning d|Lösning 2.3:1d}}
+
</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}}
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===Övning 2.3:2===
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===Exercise 2.3:2===
<div class="ovning">
<div class="ovning">
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L&ouml;s f&ouml;ljande andragradsekvationer med kvadratkomplettering
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Solve the following second order equations by completing the square
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 40: Line 40:
|width="33%" | <math>3x^2-10x+8=0</math>
|width="33%" | <math>3x^2-10x+8=0</math>
|}
|}
-
</div>{{#NAVCONTENT:Svar|Svar 2.3:2|Lösning a|Lösning 2.3:2a|Lösning b|Lösning 2.3:2b|Lösning c|Lösning 2.3:2c|Lösning d|Lösning 2.3:2d|Lösning e|Lösning 2.3:2e|Lösning 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}}
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===Övning 2.3:3===
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===Exercise 2.3:3===
<div class="ovning">
<div class="ovning">
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L&ouml;s f&ouml;ljande ekvationer direkt
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Solve the following equations directly
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 61: Line 61:
|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:Svar|Svar 2.3:3|Lösning a|Lösning 2.3:3a|Lösning b|Lösning 2.3:3b|Lösning c|Lösning 2.3:3c|Lösning d|Lösning 2.3:3d|Lösning e|Lösning 2.3:3e|Lösning 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}}
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===Övning 2.3:4===
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===Exercise 2.3:4===
<div class="ovning">
<div class="ovning">
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Best&auml;m en andragradsekvation som har r&ouml;tterna
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Find a second-degree equation which has roots
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | <math>-1\ </math> och <math>\ 2</math>
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|width="100%" | <math>-1\ </math> and <math>\ 2</math>
|-
|-
|b)
|b)
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|width="100" | <math>1+\sqrt{3}\ </math> och <math>\ 1-\sqrt{3}</math>
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|width="100" | <math>1+\sqrt{3}\ </math> and <math>\ 1-\sqrt{3}</math>
|-
|-
|c)
|c)
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|width="100" | <math>3\ </math> och <math>\ \sqrt{3}</math>
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|width="100" | <math>3\ </math> and <math>\ \sqrt{3}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.3:4|Lösning a|Lösning 2.3:4a|Lösning b|Lösning 2.3:4b|Lösning 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}}
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===Övning 2.3:5===
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===Exercise 2.3:5===
<div class="ovning">
<div class="ovning">
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
-
|width="100%" | Best&auml;m en andragradsekvation som bara har <math>\,-7\,</math> som rot.
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|width="100%" | Find a second-degree equation which only has <math>\,-7\,</math> as a root.
|-
|-
|b)
|b)
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|width="100" | Best&auml;m ett v&auml;rde p&aring; <math>\,x\,</math> som g&ouml;r att uttrycket <math>\,4x^2-28x+48\,</math> &auml;r negativt.
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|width="100" | Determine a value of <math>\,x\,</math> which makes the expression <math>\,4x^2-28x+48\,</math> negative.
|-
|-
|c)
|c)
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|width="100" | Ekvationen <math>\,x^2+4x+b=0\,</math> har en rot <math>\,x=1\,</math>. Best&auml;m v&auml;rdet p&aring; konstanten <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:Svar|Svar 2.3:5|Lösning a|Lösning 2.3:5a|Lösning b|Lösning 2.3:5b|Lösning 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}}
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===Övning 2.3:6===
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===Exercise 2.3:6===
<div class="ovning">
<div class="ovning">
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Best&auml;m det minsta v&auml;rde som f&ouml;ljande polynom antar
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Determine the smallest value that the following polynomials can take
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 101: Line 101:
|width="33%" | <math>x^2-4x+2</math>
|width="33%" | <math>x^2-4x+2</math>
|c)
|c)
-
|width="33%" | <math>x^2-5x+7</math>
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|width="33%" | <math>x^2-5x+7</math>.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.3:6|Lösning a|Lösning 2.3:6a|Lösning b|Lösning 2.3:6b|Lösning 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===
 +
<div class="ovning">
 +
Determine the largest value that the following polynomials can take
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%" | <math>1-x^2</math>
 +
|b)
 +
|width="33%" | <math>-x^2+3x-4</math>
 +
|c)
 +
|width="33%" | <math>x^2+x+1</math>.
 +
|}
 +
</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===
 +
<div class="ovning">
 +
Sketch the graph of the following functions
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%" | <math>f(x)=x^2+1</math>
 +
|b)
 +
|width="33%" | <math>f(x)=(x-1)^2+2</math>
 +
|c)
 +
|width="33%" | <math>f(x)=x^2-6x+11</math>.
 +
|}
 +
</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===
 +
<div class="ovning">
 +
Find all the points where the following curves intersect the <math>x</math>-axis.
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%" | <math>y=x^2-1</math>
 +
|b)
 +
|width="33%" | <math>y=x^2-5x+6</math>
 +
|c)
 +
|width="33%" | <math>y=3x^2-12x+9</math>
 +
|}
 +
</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===
 +
<div class="ovning">
 +
In the ''xy''-plane, shade in the area whose coordinates <math>\,(x,y)\,</math> satisfy
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="50%" | <math>y \geq x^2\ </math> and <math>\ y \leq 1 </math>
 +
|b)
 +
|width="50%" | <math>y \leq 1-x^2\ </math> and <math>\ x \geq 2y-3 </math>
 +
|-
 +
|c)
 +
|width="50%" | <math>1 \geq x \geq y^2</math>
 +
|d)
 +
|width="50%" | <math>x^2 \leq y \leq x </math>
 +
 
 +
|}
 +
</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) \displaystyle -1\ and \displaystyle \ 2
b) \displaystyle 1+\sqrt{3}\ and \displaystyle \ 1-\sqrt{3}
c) \displaystyle 3\ and \displaystyle \ \sqrt{3}

Answer | Solution a | Solution b | Solution c

Exercise 2.3:5

a) Find a second-degree equation which only has \displaystyle \,-7\, as a root.
b) Determine a value of \displaystyle \,x\, which makes the expression \displaystyle \,4x^2-28x+48\, negative.
c) The equation \displaystyle \,x^2+4x+b=0\, has one root at \displaystyle \,x=1\,. Determine the value of the constant \displaystyle \,b\,.

Answer | Solution a | Solution b | Solution c

Exercise 2.3:6

Determine the smallest value that the following polynomials can take

a) \displaystyle x^2-2x+1 b) \displaystyle x^2-4x+2 c) \displaystyle x^2-5x+7.

Answer | Solution a | Solution b | Solution c


Exercise 2.3:7

Determine the largest value that the following polynomials can take

a) \displaystyle 1-x^2 b) \displaystyle -x^2+3x-4 c) \displaystyle x^2+x+1.

Answer | Solution a | Solution b | Solution c

Exercise 2.3:8

Sketch the graph of the following functions

a) \displaystyle f(x)=x^2+1 b) \displaystyle f(x)=(x-1)^2+2 c) \displaystyle f(x)=x^2-6x+11.

Answer | Solution a | Solution b | Solution c

Exercise 2.3:9

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

a) \displaystyle y=x^2-1 b) \displaystyle y=x^2-5x+6 c) \displaystyle y=3x^2-12x+9

Answer | Solution a | Solution b | Solution c

Exercise 2.3:10

In the xy-plane, shade in the area whose coordinates \displaystyle \,(x,y)\, satisfy

a) \displaystyle y \geq x^2\ and \displaystyle \ y \leq 1 b) \displaystyle y \leq 1-x^2\ and \displaystyle \ x \geq 2y-3
c) \displaystyle 1 \geq x \geq y^2 d) \displaystyle x^2 \leq y \leq x

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

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