a) [tex]x= \sqrt{24-2x} [/tex] ⇒ [tex] x^{2} [/tex] = 24-2x⇒[tex] x^{2} [/tex]+2x-24 = 0⇒ Δ=4+96=100
[tex] x_{1}= \frac{-b+ \sqrt{delta} }{2a} = \frac{-2+10}{2} = \frac{8}{2}=4 [/tex]
[tex] x_{2}= \frac{-b- \sqrt{delta} }{2a}= \frac{-2-10}{2}= \frac{-12}{2}=-6 [/tex]
b) [tex] \sqrt{6+5x}=x [/tex] ⇒6+5x=x²⇒x²-5x-6=0 ⇒ Δ=25+24 =69⇒[tex] x_{1}= \frac{5+7}{2}= \frac{12}{2}=6
x_{2}= \frac{5-7}{2}= \frac{-2}{2}=-1 [/tex]
c) x+6=[tex] \sqrt{12+2x} [/tex] ⇒ x²+12x+36 = 12+2x ⇒ x²+10x+24 = 0 ⇒Δ= 100-96 = 4 ⇒[tex] x_{1}= \frac{-10+2}{2} = \frac{-8}{2}= -4
x_{2}= \frac{-10-2}{2}= \frac{-12}{2}=-6 [/tex]
d) 3-x = [tex] \sqrt{3x+1} [/tex] ⇒ 9-6x+x²=3x+1 ⇒x²-9x+8 = 0 ⇒ Δ=81-32 = 49
x₁=[tex] \frac{9+7}{2}= \frac{16}{2} =8
x_{2}= \frac{9-7}{2}= 1 [/tex]
e) [tex] \sqrt{ x^{2} -3x} = x-3 [/tex]⇒ x²-3x = x²-6x+9 ⇒ 3x-9 = 0 ⇒3x = 9 ⇒x=[tex] \frac{1}{3} [/tex]
f) [tex] \sqrt{10x-6 x^{2} } [/tex]=3x-1 ⇒10x-6x²= 9x²-6x+1 ⇒ 15x²-16x+1 =0 ⇒ Δ= 256-60= 196[tex] x_{1}= \frac{16+14}{30}= \frac{30}{30}=1
x_{2} = \frac{16-14}{30}= \frac{2}{30}= \frac{1}{15} [/tex]
