2.3 Exercises

From Förberedande kurs i matematik 1

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===Exercise 2.3:4===
===Exercise 2.3:4===
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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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===Exercise 2.3:6===
===Exercise 2.3:6===
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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
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|width="33%" | <math>x^2-4x+2</math>
|width="33%" | <math>x^2-4x+2</math>
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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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</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}}
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===Exercise 2.3:7===
===Exercise 2.3:7===
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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
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|width="33%" | <math>-x^2+3x-4</math>
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|width="33%" | <math>x^2+x+1</math>
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|width="33%" | <math>f(x)=(x-1)^2+2</math>
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|width="33%" | <math>f(x)=x^2-6x+11</math>
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===Exercise 2.3:9===
===Exercise 2.3:9===
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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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===Exercise 2.3:10===
===Exercise 2.3:10===
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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, plot all the points whose coordinates <math>\,(x,y)\,</math> satisfy
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Revision as of 23:32, 11 November 2008

       Theory          Exercises      

Exercise 2.3:1

Complete the square of the expressions

a) \displaystyle x^2-2x b) \displaystyle x^2+2x-1 c) \displaystyle 5+2x-x^2 d) \displaystyle x^2+5x+3
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

Solve the following second order equations by completing the square

a) \displaystyle x^2-4x+3=0 b) \displaystyle y^2+2y-15=0 c) \displaystyle y^2+3y+4=0
d) \displaystyle 4x^2-28x+13=0 e) \displaystyle 5x^2+2x-3=0 f) \displaystyle 3x^2-10x+8=0
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

Solve the following equations directly

a) \displaystyle x(x+3)=0 b) \displaystyle (x-3)(x+5)=0
c) \displaystyle 5(3x-2)(x+8)=0 d) \displaystyle x(x+3)-x(2x-9)=0
e) \displaystyle (x+3)(x-1)-(x+3)(2x-9)=0 f) \displaystyle x(x^2-2x)+x(2-x)=0
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

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
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

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
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

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
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

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
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

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
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

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
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

In the xy-plane, plot all the points 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
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
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