Question

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Answer 1

1.a) x^2=1
x=√1
x=+/- 1
b) 2x^2= -8
x^2= -4
x= -√4
x= -2

2.34+15(x+3)^2=169
15(x+3)^2=135
(x+3)^2=9
x+3=√9
x+3=3
x=0 sau
x+3= -3
x= -6

3.9+6x+x^2+9-6x+x^2=36
18+2x^2=36
2x^2=18
x^2=9
x=+/- √9
x= +/- 3 deci poate fi soluţia a= -3

4.x^2+x^2+4x+4+x^2+8x+16=12x+32
3x^2+12x+20-12x=32
3x^2+20=32
3x^2=12
x^2=4
x=√4
x=+/- 2

5.9x^2+24x+16=4x^2+4x+1
9x^2-4x^2+24x-4x+16-1=0
5x^2+20x+15=0
delta=b^2-4ac=400-300=100
x1= (-b+√delta)/2a=(-20+10)/10= -10/10= -1
x2= (-b-√delta)/2a=(-20-10)/10= -30/10= -3

Answer 2

$1)a) {x}^{2} = 1$

$x = \pm \sqrt{1}$

$x = \pm1$

$b)2 {x}^{2} + 1 = - 7$

$2 {x}^{2} = 7 + 1$

$2 {x}^{2} = 8 \: | \div 2$

${x}^{2} = 4$

$x = \pm \sqrt{4}$

$x = \pm2$

$c) {x}^{2} + 2 {x}^{2} = 4 {x}^{2}$

${x}^{2} + 2 {x}^{2} - 4 {x}^{2} = 0$

$- {x}^{2} = 0$

${x}^{2} = 0$

$x = 0$

$2)34 + 15 {(x + 3)}^{2} = {( - 13)}^{2}$

$34 + 15( {x}^{2} + 6x + 9) = 169$

$34 + 15 {x}^{2} + 90x + 135 = 169$

$15 {x}^{2} + 90x + 34 + 135 - 169 = 0$

$15 {x}^{2} + 90x + 169 - 169 = 0$

$15 {x}^{2} + 90x + 0 = 0$

$15 {x}^{2} + 90x = 0$

$a = 15$

$b = 90$

$c = 0$

$\Delta = {b}^{2} - 4ac$

$\Delta = {90}^{2} - 4 \times 15 \times 0$

$\Delta = 8100>0=>\:\exists\:x_{1}\:\neq\:x_{2}$

$x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$x_{1,2}=\frac{-90\pm\sqrt{8100}}{2 \times 15}$

$x_{1,2}=\frac{-90\pm90}{30}$

$x_{1}=\frac{-90 + 90}{30}$

$x_{1}=\frac{0}{30}$

$x_{1}=0$

$x_{2}=\frac{-90 - 90}{30}$

$x_{2}=\frac{-180}{30}$

$x_{2}= - 6$

$3) {( 3 + x)}^{2} + {(3 - x)}^{2} = 36$

${[ 3+ ( - 3)]}^{2} + {[3 - ( - 3)]}^{2} = 36$

${(3 - 3)}^{2} + {(3 + 3)}^{2} = 36$

${0}^{2} + {6}^{2} = 36$

$0 + 36 = 36$

$36 = 36 \: (A)$

$=>a=-3\:este\:solutie\:a\:ecuatiei$

$4) {x}^{2} + {(x + 2)}^{2} + {(x + 4)}^{2} = 4(3x + 8)$

${x}^{2} + {x}^{2} + 4x + 4 + {x}^{2} + 8x + 16 = 12x + 32$

${x}^{2} + {x}^{2} + {x}^{2} + 4x + 8x - 12x = 32-16-4$

$3 {x}^{2} = 12\:|\div3$

${x}^{2} = 4$

$x= \pm\:\sqrt{4}$

$x = \pm\:2$

$5) {(3x + 4)}^{2} = {(2x + 1)}^{2}$

$9 {x}^{2} + 24x + 16 = 4 {x}^{2} + 4x + 1$

$9 {x}^{2} - 4 {x}^{2} + 24x - 4x + 16 - 1 = 0$

$5 {x}^{2} + 20x + 15 = 0$

$a = 5$

$b = 20$

$c = 15$

$\Delta = {b}^{2} - 4ac$

$\Delta = {20}^{2} - 4 \times 5 \times 15$

$\Delta = 400 - 300$

$\Delta = 100 > 0 = > \exists \: x_{1} \: \neq \: x_{2}$

$x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$x_{1,2}=\frac{-20\pm\sqrt{100}}{2 \times 5}$

$x_{1,2}=\frac{-20\pm10}{10}$

$x_{1}=\frac{-20 + 10}{10}$

$x_{1}=\frac{ - 10}{10}$

$x_{1}= - 1$

$x_{2}=\frac{ - 20 - 10}{10}$

$x_{2}=\frac{ - 30}{10}$

$x_{2}= - 3$

$6) {x}^{2} + 6x + 8 = 6 + 4 \sqrt{3}$

${x}^{2} + 6x + 8 - 6 - 4 \sqrt{3} = 0$

${x}^{2} + 6x + 2 - 4 \sqrt{3} = 0$

$a = 1$

$b = 6$

$c = 2 - 4 \sqrt{3}$

$\Delta = {b}^{2} - 4ac$

$\Delta = {6}^{2} - 4 \times 1 \times (2 - 4 \sqrt{3} )$

$\Delta = 36 + 8 - 16 \sqrt{3}$

$\Delta = 44 - 16 \sqrt{3}>0=>\:\exists \:x_{1}\:\neq\:x_{2}$

$x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$x_{1,2}=\frac{-6\pm\sqrt{44 - 16 \sqrt{3} }}{2 \times 1}$

$x_{1,2}=\frac{-6\pm\sqrt{44 - 16 \sqrt{3} }}{2}$

$x_{1}=\frac{-6 + \sqrt{44 - 16 \sqrt{3} }}{2}$

$x_{2}=\frac{-6 - \sqrt{44 - 16 \sqrt{3} }}{2}$