Odpowiedź:
zad. 5
1)
[tex]a) \: 3 {cm}^{2} = x \: ( x + 2) \\ b)1.5 {cm}^{2} = x( \frac{1}{2} x - 1) \\ c)1 {m}^{2} = (x - 1 {)}^{2} [/tex]
2)
[tex]a)3 = x(x + 2) \\ 3 = {x}^{2} + 2x \\ {x}^{2} + 2x - 3 = 0 \\ a = 1 \\ b = 2 \\ c = - 3[/tex]
[tex]b)1.5 = x( \frac{1}{2}x - 1) \\ 1.5 = \frac{1}{2} {x}^{2} - x \\ \frac{1}{2} {x}^{2} - x - 1.5 = 0 \\ a = \frac{1}{2} \\ b = - 1 \\ c = - 1.5[/tex]
[tex]c)1 = (x - 1 {)}^{2} \\ 1 = {x}^{2} - 2x + 1 \\ {x}^{2} - 2x = 0 \\ a = 1 \\ b = - 2 \\ c = 0[/tex]
3)
a)
[tex]( - 3 {)}^{2} + 2 \times ( - 3) - 3 = 0 \\ 9 - 6 - 3 = 0 \\ 0 = 0 \\ [/tex]
należy[tex]( - 2 {)}^{2} + 2( - 2) - 3 = 0 \\ 4 - 4 - 3 = 0 \\ - 3 = 0[/tex]
nie należy
[tex]( - 1 {)}^{2} + 2( - 1) - 3 = 0 \\ 1 - 2 - 3 = 0 \\ - 4 = 0[/tex]
nie należy
[tex] {0}^{2} + 2 \times 0 - 3 = 0 \\ 0 + 0 - 3 = 0 \\ - 3 = 0[/tex]
nie należy
[tex] {1}^{2} + 2 \times 1 - 3 = 0 \\ 1 + 2 - 3 = 0 \\ 3 - 3 = 0 \\ 0 = 0[/tex]
należy
[tex] {2}^{2} + 2 \times 2 - 3 = 0 \\ 4 + 4 - 3 = 0 \\ 8 - 3 = 0 \\ 5 = 0[/tex]
nie należy
[tex] {3}^{2} + 2 \times 3 - 3 = 0 \\ 9 + 6 - 3 = 0 \\ 12 = 0[/tex]
b)
[tex] \frac{1}{2} ( - 3 {)}^{2} - ( - 3) - 1.5 = 0 \\ \frac{1}{2} \times 9 + 3 - 1.5 = 0 \\ 6 = 0[/tex]
nie należy[tex] \frac{1}{2} ( - 2 {)}^{2} - ( - 2) - 1.5 = 0 \\ \frac{1}{2} \times 4 + 2 - 1.5 = 0 \\ 2 + 2 - 1.5 = 0 \\ 4 - 1.5 = 0 \\ 2.5 = 0[/tex]
nie należy
[tex] \frac{1}{2} \times ( - 1) - ( - 1) - 1.5 = 0 \\ - 0.5 + 1 - 1.5 = 0 \\ - 1 = 0[/tex]
nie należy
[tex] \frac{1}{2} \times {0}^{2} - 0 - 1.5 = 0 \\ 0 - 0 - 1.5 = 0 \\ - 1.5 = 0[/tex]
nie należy
[tex] \frac{1}{2} \times {1}^{2} - 1 - 1.5 = 0 \\ 0.5 - 1 - 1.5 = 0 \\ - 2 = 0[/tex]
nie należy
[tex] \frac{1}{2} \times {2}^{2} - 2 - 1.5 = \frac{1}{2} \times 4 - 2 - 1.5 = 0 \\ 2 - 3.5 = 0 \\ - 1.5 = 0[/tex]
nie należy
[tex] \frac{1}{2} \times {3}^{2} - 3 - 1.5 = 0 \\ 4.5 - 4.5 = 0 \\ 0 = 0[/tex]
należy
c)
[tex]( - 3 {)}^{2} - ( - 3) = 0 \\ 9 + 3= 0 \\ 12= 0[/tex]
nie należy
[tex]( - 2 {)}^{2} - 2 \times ( - 2) + 1 = 0 \\ 4 + 4 = 0 \\ 8 = 0[/tex]
nie należy
[tex]( - 1 {)}^{2} - 2( - 1) = 0 \\ 1 + 2 = 0 \\ 3 = 0[/tex]
nie należy
[tex] {0}^{2} - 2 \times 0 = 0 \\ 0 = 0[/tex]
należy
[tex] {1}^{2} - 2 \times 1 = 0 \\ 1 - 2 =0 \\ - 1 = 0[/tex]
nie należy
[tex] {2}^{2} - 2 \times 2= 0 \\ 4 - 4 = 0 \\ 0 = 0[/tex]
należy
[tex] {3}^{2} - 2 \times 3 = 0 \\ 9 - 6 = 0 \\ 3 = 0[/tex]
nie należy
4)
[tex]a) {x}^{2} + 2x - 3 = 0 \\ ∆ = {b}^{2} - 4ac = {2}^{2} - 4 \times ( - 3) \times 1 = 4 + 12 = 16 \\ \sqrt{∆} = \sqrt{16} = 4 \\ x_{1} = \frac{ - b - \sqrt{∆} }{2a} = \frac{ - 2 - 4}{2} = \frac{ - 6}{2} = - 3 \\ x_{2} = \frac{ - b + \sqrt{∆} }{2a} = \frac{ - 2 + 4}{2} = \frac{2}{2} = 1[/tex]
x>0
x=1 cm
x+2=1+2=3 cm
[tex]b) \frac{1}{2} {x}^{2} - x - 1.5 = 0 \\ ∆ = {b}^{2} -4ac = ( - 1 {)}^{2} - 4 \times ( \frac{1}{2}) \times ( - 1.5) = 1 + 3 = 4 \\ \sqrt{∆} = \sqrt{4} = 2 \\ x_{1} = \frac{ - b - \sqrt{∆} }{2a} = \frac{ 1 - 2}{1} = - 1 \\ x_{2} = \frac{ - b + \sqrt{∆} }{2a} = \frac{1 + 2}{1} = 3[/tex]
x>0
x=3 cm
[tex] \frac{1}{2}x - 1 = \frac{1}{2} \times 3 - 1 = 1.5 - 1 = 0.5 \: cm[/tex]
[tex]c) {x}^{2} - 2x = 0 \\ ∆ = {b}^{2} - 4ac = ( - 2 {)}^{2} - 4 \times 1 \times 0 = 4 - 0 = 4 \\ \sqrt{∆} = \sqrt{4} = 2 \\ x_{1} = \frac{ - b - \sqrt{∆} }{2a} = \frac{2 - 2}{2} = 0 \\ x_{2} = \frac{2 + 2}{2} = \frac{4}{2} = 2[/tex]
x>0
x=2
x-1=2-1=1 cm
