2.3 Exercises
From Förberedande kurs i matematik 1
| 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 |
Svar
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Lösning a
Lösning b
Lösning c
Lösning d
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 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Lösning f
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 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Lösning f
Exercise 2.3:4
Determine 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} |
Exercise 2.3:5
| a) | Determine 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\, be 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\,. |
Exercise 2.3:6
Determine the smallest value that the following polynomial can take
| a) | \displaystyle x^2-2x+1 | b) | \displaystyle x^2-4x+2 | c) | \displaystyle x^2-5x+7 |
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 |
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 |
Exercise 2.3:9
Find all the points where the x-axis and the following curves intersect.
| a) | \displaystyle y=x^2-1 | b) | \displaystyle y=x^2-5x+6 | c) | \displaystyle y=3x^2-12x+9 |
Exercise 2.3:10
In the xy-plane, draw in all the points whose coordinates \displaystyle \,(x,y)\, satisfy
| a) | \displaystyle y \geq x^2\ och \displaystyle \ y \leq 1 | b) | \displaystyle y \leq 1-x^2\ och \displaystyle \ x \geq 2y-3 |
| c) | \displaystyle 1 \geq x \geq y^2 | d) | \displaystyle x^2 \leq y \leq x |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
