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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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|width="100%" | Determine a second-degree equation which only has <math>\,-7\,</math> as a root.
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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>
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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>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, shade in the area whose coordinates <math>\,(x,y)\,</math> satisfy
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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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